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\begin{document}
\bibliographystyle{unsrt}
\title{Quantum Decoherence and the Approach to
Equilibrium (Part I)}
\author{Meir Hemmo\thanks{Department of Philosophy,
University of Haifa, Haifa 31905, Israel.
(email: meir@research.haifa.ac.il)} , Orly Shenker\thanks{Department
of Philosophy, Logic and Scientific Method,
London School of Economics, London WC2A 2AE,
UK. (email: o.shenker@lse.ac.uk)}}
\maketitle
\begin{abstract}
\noindent
We discuss a recent proposal by Albert (1994a,b; 2000, Chapter 7) to
recover thermodynamics on a purely dynamical basis, using the quantum
theory of the collapse of the
wave function of Ghirardi, Rimini and Weber (1986). We
propose an alternative way to explain thermodynamics within no-collapse
interpretations of quantum mechanics. Our approach relies on the
standard quantum mechanical models of environmental decoherence
of open systems, \eg Joos and Zeh (1985) and Zurek and Paz (1994).
This paper presents the two approaches and discusses their advantages. The
problems they face will be discussed in a sequel (Hemmo and Shenker 2002b).
\end{abstract}
{\em Keywords}: collapse; decoherence; entropy; quantum mechanics;
thermodynamics.
\section{Introduction}\label{intro}
Our experience tells us that macroscopic thermodynamic
systems invariably evolve towards high entropy states in an irreversible
way. A central problem in the foundations of statistical mechanics
(both classical and quantum) is to explain this experience by appealing
to the underlying dynamics {\em only}. The aim here is twofold. First,
to explain the macroscopic thermodynamic phenomena\footnote{By
expressions such as `thermodynamic phenomena', `thermodynamic
(law-like) regularities', and similar ones we mean the results of
measurements as predicted by the zeroth and second laws of
thermodynamics (stating the spontaneous and irreversible approach to equilibrium and entropy increase.} on the basis of the
dynamical equations of motion that operate at the microscopic
level, possibly using some probabilistic hypotheses. Then, to
justify those probabilistic
hypotheses by the same underlying dynamics. Hitherto neither of the two
aims has been satisfactorily accomplished.\footnote{See overviews of
this problem in Sklar (1993) and Guttmann (1999).}
Albert (1994a,b; 2000, Chapter 7) has proposed to solve these
problems by appealing to the theory of the collapse of the quantum state
by Ghirardi, Rimini and Weber (GRW) (1986).\footnote{In this paper we
only consider the extent to which the GRW theory can be successful in the
recovery of thermodynamics. This theory faces some serious problem, some
of which are mentioned in section \ref{grw} below.}
In this paper we propose an alternative
way of explaining the laws of thermodynamics, in particular the approach to
equilibrium and the increase of entropy, using the quantum
mechanical dynamics in {\em no-collapse} theories. Our proposal finds
support in results in decoherence theory which strongly suggest
that interactions with the environment are crucial for the
emergence of quasi-classical and thermodynamic behaviour. We use the
standard models of so-called environmental decoherence of open
systems (see Zurek (1982, 1993), Caldeira and Leggett (1983),
Joos and Zeh (1985), Giulini et. al. (1996) and references therein),
and recent results about the evolution of the von Neumann entropy of open
(decohering) systems by Zurek, Paz and Habib (1993),
Zurek and Paz (1994, 1995); see also Paz and Zurek (1999, Chap. 6, pp.
55-65)). This paper, however, while focusing on quantum mechanics without
collapse, does not defend any specific no-collapse interpretation of
quantum mechanics (\eg pilot-wave, many worlds or modal
theories\footnote{See DeWitt and Graham (1973) on many worlds; Healey and
Hellman (1998) and Dieks and Vermaas (1998) on modal theories; Cushing,
Goldstein and Fine (1996) and Bub (1997) on the pilot-wave theory.}).
The role of decoherence in the recovery of classical
mechanics and of thermodynamics has been investigated by other authors,
see in particular Zeh (1992, Chapter 4, section 4.2.2) and Wallace (2001).
The present paper follows a more condensed argument given
in Hemmo and Shenker (2001).
The paper is structured as follows.
In section \ref{setup} we present the problem of justifying the
thermodynamic regularities in
classical statistical mechanics. In section \ref{grw} we turn to the
quantum mechanical context and we discuss Albert's approach to the problem
using the GRW theory of the collapse of the quantum state. In section
\ref{dec} we give a brief description of the standard model of environmental
decoherence, and we describe recent results by Zurek, Habib and Paz
concerning the connection between decoherence of open systems and the
evolution of the von Neumann entropy. In section \ref{exp} we present our
approach to the problem in which we make use of both environmental
decoherence and the induced dynamics of open systems in no-collapse
interpretations of quantum mechanics. In section \ref{stoch} we consider
the role of probabilities and stochasticity in Albert's GRW approach
and in no-collapse approach. In a
sequel to this paper (Hemmo and Shenker 2002b; hereafter HS2002b) we
address further problems faced by the two approaches.
\section{The Problem}\label{setup}
Empirical evidence suggests that
irreversibility and the approach to
equilibrium are universal (for systems that are isolated and
contained in a finite volume).\footnote{In this paper we use terms like
`the second law', `law of entropy increase', `principle of approach to
equilibrium', etc. interchangeably. The exact meaning of the second law of
thermodynamics is, however, not clear; see Uffink (2001). It is even an
open question whether the second law entails or assumes a time asymmetric
spontaneous evolution to equilibrium; see Brown and Uffink (2001).}
Systems evolve to equilibrium invariably and
irrespective of their initial conditions. These ideas for the heart of
thermodynamics. Can the universal approach to
equilibrium be explained on the basis of the
underlying dynamics?
To illustrate the problem consider the following example.
A gas cloud is confined to the left hand side of a
container by a partition, which is removed at time $t_0$.
The principles of thermodynamics dictate that
the gas will evolve to equilibrium, that is, expand and fill out
the container, and remain in the expanded state indefinitely.
The universal phenomenon of the approach to equilibrium is
classically understood as the macroscopic appearance of
occurrences at the microscopic level.
This general idea is applicable only in the right
circumstances, and these ought to be taken into account.
First, in systems characterised by a small number of degrees of
freedom, fluctuations (which disagree with the predictions of
thermodynamics) are dominant.
The law-likeness of thermodynamics, in particular the approach to equilibrium, emerges only when we move to
macroscopic systems with many degrees of freedom.
Second, where quantum mechanical phenomena like superpositions
dominate, thermodynamic magnitudes and their evolution are not
always well-defined. In fact, we take it that
without solving the measurement problem quantum mechanics has no
empirical content at all, thermodynamic or otherwise. For this reason,
the explanation of thermodynamic behaviour within a quantum
mechanical setting crucially depends
on the way the measurement problem is solved in quantum
mechanics.
Hence, if one wishes to explain the thermodynamic regularities
on the basis of quantum mechanics, one has to consider
an interpretation of quantum mechanics in which
the measurement problem is solved.
These are the circumstances on which both approaches discussed in this paper focus.
One problem in explaining macroscopic occurrences on the basis of the
classical microscopic dynamics is that the latter allows for
micro evolutions that (would) appear at the macroscopic level as
anti-thermodynamic (had they
occurred), such as the gas remaining in the left hand side of the
container, say ten minutes from now. We call micro evolutions, and
the microstates along them, {\em thermodynamic normal} if the
regularities they exhibit correspond to the laws of thermodynamics
(in particular the second law) for a suitable time interval $T$. (A time interval is suitable if it is long enough in thermodynamic time scales but short enough
so that Poincar\`e recurrence is unlikely to occur).
Micro evolutions (and the microstates along them) which don't
satisfy this condition
will be called thermodynamic {\em abnormal}.\footnote{These
definitions are in agreement with Albert (1994a,b; 2000).}
The existence of thermodynamic abnormal states,
as predicted by the underlying classical
dynamics, contradicts the letter of the second law
of thermodynamics. This situation has been known for a long time:
J. C. Maxwell had proposed his famous Demon to illustrate it (see Earman and Norton 1998).
In classical statistical mechanics there are two main
grand schools to solving this problem: following
Boltzmann and following Gibbs.
In the Boltzmann school the properties of a system (including its entropy) are taken
to be properties of the system's microstate
through its relation to the system's macrostate.
Among the initial microstates there are abnormal
ones, namely states that lead to anti thermodynamic
evolutions. The problem now is to explain why, despite the {\em possibility} of such anti-thermodynamic evolutions, the {\em actual} world obeys the laws of thermodynamics.
\footnote{See overviews of this problem in the classical
context in Sklar (1993), Guttmann (1999), Albert (2000).} One solution is to argue that {\em as
a matter of fact} the actual initial state and
evolution are normal. This is a matter of {\em fact},
not of {\em law}, and therefore it does not need to be explained
beyond its mere stipulation.\footnote{See, for example,
Sklar (1973) p. 210.} In order to derive the laws of
thermodynamics we must
add this fact to the underlying dynamics (call this the
{\em matter-of-fact} approach)
A second approach (within the Boltzmannian school) seeks to give some explanation
for this fact by
stipulating a probability distribution over the
possible initial states of the universe. This
distribution entails that most initial states evolve (with
certainty) in accordance with the thermodynamic laws, and the
minority of initial states evolve (again, with certainty)
anti-thermodynamically. This approach goes on to assume that the actual
initial state of the universe belongs to the majority. It is {\em
typical}. (Call this the {\em typicality} approach.) Like the
matter-of-fact approach, the typicality approach also needs to
postulate something beyond the underlying dynamics in order to
recover or explain thermodynamics. The matter-of-fact approach
postulates the actual initial state, and the
typicality approach postulates the initial probability distribution
and the typicality of the actual initial state. \footnote{A typicality
approach is advocated by the Bohmian school in the interpretation of
quantum mechanics in the context of justifying the quantum mechanical
probability distribution $\abs{\ket{\Psi(x,t)}}^2$ over
the positions in Bohm's theory, \eg Goldstein's proposal;
see Bricmont et.al. (2001).}
Albert (2000, Chapter 4), recently proposed a third solution to this
problem, again - in a Boltzmannian framework. Albert hypothesises that
{\em all} the physically possible initial states of the universe have
been thermodynamic normal. This leads to a rejection of the standard
understanding of a uniform distribution over the microstates
corresponding to the {\em present} macrostate.
In all three approaches the additional
non-dynamical postulates are extremely
hard to justify.
The Gibbsian school, by contrast, maintains that the properties of
individual systems correspond to ensemble averages, and so the
effect of the relatively small number of thermodynamic abnormal
trajectories is negligible. The average behaves like the majority, namely,
in accordance with the thermodynamic laws. Since the average is said to
correspond to measurable
quantities pertaining to individual systems, this approach needs to prove
the uniqueness of the probability measure which is the basis for calculating
the averages. A satisfactory proof based on the underlying dynamics has
not yet been found.\footnote{See Sklar (1993) and Guttmann (1999) on this
problem.} It is known that one can obtain this result whenever the
system is ergodic, but ergodicity may not be sufficient nor necessary to explain thermodynamics. Not sufficient, since it does not provide predictions for finite time intervals. Not necessary, since some interesting and relevant systems may not be ergodic, by KAM's
theorem (see Walker and Ford 1969).\footnote{For problems
in the ergodic approach, see Earman and Redei (1996).}
And so also the Gibbsian school relies on non-dynamical postulates.
\footnote{In this school there are additional problems. One is explaining why phase averages yield predictions regarding individual systems. Another is that, since entropy is a property of the probability distribution, it does not change at all. To solve this problem Gibbs devised the idea of coarse graining, which is problematic; see Ridderbos (2002).}
Until recently, no approach to the foundations of statistical mechanics has
been able to overcome these difficulties and to rely on the dynamics only in
recovering thermodynamics. Albert's (1994a,b and 2000; Chapter 7) recent
proposal (on which we shall focus here) attempts to
provide a way to do precisely this, by taking seriously
the fact that the underlying dynamics is quantum mechanical,
rather than classical. Albert's proposal belongs to the
Boltzmannian school in that it takes entropy to be
a property of the microstates of individual systems, rather than a
property of ensembles or probability distributions over microstates as in
the Gibbsian approach. Therefore he focuses on the dynamical
evolution of individual systems, rather than on the evolution
of ensembles or probability distributions. It seems to us possible to
apply Albert's ideas in a Gibbsian framework as well; we do not
undertake this here.
\section{GRW Jumps and Thermodynamics}\label{grw}
Albert({\em ibid}) proposes to explain the thermodynamic regularities
by relying solely on the stochastic dynamics of the quantum
state as prescribed by the quantum theory of the collapse of the
wave function proposed by Ghirardi, Rimini and Weber (1986).
On his approach it is an intrinsic feature of the
GRW dynamics of the quantum state that {\em every single one} of the
possible initial microstates of a thermodynamic system has a high
probability to evolve to states which are compatible with the predictions
of thermodynamics, and therefore there is no need to add any of the non-dynamic postulates used in the classical case, as described above. As we shall see Albert's
approach relies heavily on the fact that the GRW dynamics
is genuinely stochastic.
We now turn to a detailed discussion of Albert's approach. We
start (section \ref{Bell}) by briefly presenting the GRW theory and
how it solves the measurement problem in quantum mechanics. Then
(section \ref{Alb1}) we describe Albert's approach as to how to
recover the thermodynamic regularities using the GRW collapses.
Finally (section \ref{Alb2}), we consider in
more detail some features of Albert's approach.
\subsection{The GRW Theory}\label{Bell}
Albert's approach makes an explicit linkage
between the GRW solution to the measurement problem in the quantum
theory of measurement and the implications of the GRW theory
concerning the time evolution of thermodynamic systems.
The measurement problem arises in quantum mechanics
as a straightforward consequence
of applying the Schr\"odinger linear and deterministc dynamics
to the measurement interaction. This dynamics results for a generic
measurement interaction in a superposition
of the form
\begin{equation}
\ket{\Psi}=\sum_i\mu_i\ket{\psi_i}\ten\ket{\varphi_i},
\label{eq:sup}
\end{equation}
where the kets $\ket{\psi_i}$ represent some suitably defined pointer
states of the measuring apparatus (typically, the $\ket{\psi_i}$
are eigenstates of the pointer position), and the
$\ket{\varphi_i}$ are some states of the system. The problem is that
in states of the form (\ref{eq:sup}) the measurement has no definite outcome
(except in the special case where all but one of the $\mu_i$ are zero),
since the final (reduced) state of the apparatus cannot in general be
described in terms of an ensemble of systems in a classical mixture (in
which the $\abs{\mu_i}^2$ represent
the probabilities for each $\ket{\psi_i}$ to actually be the case).
The GRW theory (formulated for non relativistic quantum mechanics)
solves this problem by modifying the Schr\"odinger
linear dynamics. In particular, the Schr\"odinger equation of motion
is changed by adding to it a non-linear
and stochastic factor (so-called a {\em jump} factor).
This jump generates on occasion the collapse of
the wave function in a way that depends on the mass density
of the system (roughly, the frequencies of the collapses are
proportional to the number of the number of particles or to
the mass density of the system depending on the model).
For our purposes it is enough to present Bell's (1987)
version of the elementary and non-relativistic theory. This goes roughly as follows \footnote{For a more detailed
discussion of the GRW theory see GRW (1986), Bell (1987),
Ghirardi (2000) and references therein.}
Consider the quantum mechanical wave function of a
composite $N$ particles system:
\begin{equation}
\psi(t, {\bf r}_1, {\bf r}_2,...,{\bf r}_N).
\label{eq:psi}
\end{equation}
The time evolution of the wave function usually (at almost
all times) satisfies the deterministic Schr\"odinger equation.
But sometimes {\em at random} the wave function collapses
(these collapses are known as the GRW {\em jumps}) onto a
wave function $\psi_\ell$ localised in position which has
the (normalised) form
\begin{equation}
\psi_\ell = \frac{j({\bf x}-{\bf r}_n)
\;\psi(t, {\bf r}_1, {\bf r}_2,...,{\bf r}_N)}
{R_n({\bf x})},
\label{eq:loc}
\end{equation}
where ${\bf r}_n$ in the jump factor $j({\bf x}-{\bf r}_n)$
is randomly chosen from the arguments ${\bf r}_1,...,{\bf r}_n$
of the wave function immediately before the jump.
$R$ in (\ref{eq:loc}) is a
renormalisation factor:
\begin{equation}
\abs{R_n({\bf x})}^2 = \int {\rm d}^3 {\bf r}_1...{\rm d}^3
{\bf r}_N\abs{j\psi}^2,
\end{equation}
and the jump factor $j$ is also normalised:
\begin{equation}
\int {\rm d}^3 {\bf x}\;\abs{j({\bf x})}^2=1.
\label{eq:j}
\end{equation}
For $j$ GRW suggest the Gaussian:
\begin{equation}
j({\bf x})=K\; {\rm exp} (-{\bf x}^2/2\Delta^2),
\label{eq:gau}
\end{equation}
where the width $\Delta$ of the Gaussian is a new constant
of nature: $\Delta\approx 10^{-5}{\rm cm}$. Its size is chosen so that the
spontaneous collapses will not result in an observable violation of energy
conservation. (More on this point in (HS2002b) section 4.)
Probabilities enter the theory twice. First, the probability
that the collapsed wave function $\psi_\ell$ is centered around
the point ${\bf x}$ is given by
\begin{equation}
{\rm d}^3{\bf x}\abs{R_n({\bf x})}^2.
\label{eq:born}
\end{equation}
This probability distribution, as can be seen, is proportional to
the standard quantum mechanical probability given by the Born rule
for a position measurement on a system with a wave function
$\psi(t, {\bf r})$ just prior to the jump. Second, the probability
in for a GRW jump to take place in a unit of time is
\begin{equation}
\frac{N}{\tau},
\label{eq:ntau}
\end{equation}
where $N$ is the number of arguments in the wave function
(in Bell's model it may be interpreted as the number of
particles), and $\tau$ is another new constant of nature
($\tau\approx 10^{15}\,{\rm sec}\approx 10^8\; {\rm year}$).
Note that the expression (\ref{eq:ntau}) does not depend
on the quantum wave function, but only on $N$. This is
essentially the whole theory.
As it stands, it seems that this theory cannot be generalised to
relativistic and field theories, since the GRW jumps are applied to
particle's positions and not field variables, and the collapse
rates are determined by particle numbers.\footnote{see Pearle, Ghirardi
and Grassi (1990) and Ghirardi (2000)
on the problem of generalising the theory to the
relativistic domain.}. As part of an attempt to solve this
problem more general models are considered in which the collapse
rates are defined such that they are increased exponentially in
correspondence with the
{\em mass density} of the system.
In both models it is a straightforward consequence of the standard
quantum mechanical treatment of composite systems (in particular,
of non factorisable quantum states) that a single GRW jump of
any one of the subsystems in the composite is enough to bring
about a collapse of the global wave function.
For microscopic systems collapses have extremely low probability
to occur, so that the quantum
mechanical Schr\"odinger equation turns out to be almost
literally true at all times just as no-collapse
quantum mechanics predicts (and experiment confirms).
However, for massive macroscopic systems (or for systems with $10^{23}$
particles) collapses are highly probable at all times.
In measurement situations the GRW theory implies that
superpositions of macroscopic pointer states of the form
(\ref{eq:sup}) collapse with extremely high probability onto the
localised states $\ket{\psi_i}$ on time scales that are much
faster than measurement times. In particular, the probability
that the wave function of the composite of system plus apparatus
will stay in the superposition (\ref{eq:sup}) for more than
a fraction of a second (\ie by the time the measurement is complete)
vanishes exponentially. Moreover, whenever the wave function in
(\ref{eq:sup}) has a spatial spread which is larger than $\Delta$,
any GRW jump will result in a {\em localisation} of the wave function.
That is, the jump will reduce the wave function onto one of the terms
$\ket{\psi_i}\ten\ket{\varphi_i}$, in which the pointer is in the
localised state $\ket{\psi_i}$,
where the probability for the $i$-th term (see equation
(\ref{eq:born})) is given as usual
by the squared amplitude $\abs{\mu_i}^2$.
This means that in a sequence of quantum mechanical measurements
the GRW jumps result in definite outcomes with frequencies that are
(approximately) equal to the Born-rule probabilities $\abs{\mu_i}^2$.
The measurement problem is solved as long as measurements
involve a macroscopic recording of the result in position (\eg a moving
pointer of a measuring device, particles hitting on macroscopically
separated regions of a computer screen, etc.).
The following properties of the GRW theory will be important
later. First, the dynamics is fundamentally chancy. The time of
the collapse and the center of the Gaussian into which the wave
function collapses are determined in a purely chancy way.
Second, the time evolution resulting from the GRW dynamics
is non-invariant under time reversal. Past states cannot be retrodicted
by the GRW, not even approximately or in a probabilistic way. Third the
GRW jumps, by construction, occur only in position, and in this sense
the quantum mechanical `position basis' is given a physically
preferred status.
\subsection{Albert's Approach to Thermodynamics}\label{Alb1}
The key idea in Albert's approach is this. In the GRW theory the
jumps mean that the system actually undergoes stochastic transitions from
one state to another. In the context of the recovery of
thermodynamics it is useful to think about the GRW jumps as if they induce
{\em stochastic perturbations} of the Schr\"odinger trajectory of the
system. This means that the GRW trajectory can be seen as a patchwork of
segments of different Schr\"odinger trajectories each of which corresponds
to a different initial state of the system. The system jumps
from one Schr\"odinger trajectory to another, such that the
net result is an effective stochastic trajectory. In other words,
the system performs a random walk in the space of all possible
Schr\"odinger trajectories where the probabilities
are given by (\ref{eq:born}).
The connection to thermodynamics goes as follows.
We want to determine whether the time
evolution of a given system is thermodynamic normal or abnormal.
Consider the spreading out of a gas in the box.
When the partition is removed at
$t_0$ the composite wave function of the gas is
\begin{equation}
\ket{\Psi(0)}=\sum_i\lambda_i(0)\ket{\Phi_i},
\label{eq:gas}
\end{equation}
where the $\ket{\Phi_i}$ are some wave functions in position
representation and the $\lambda_i(0)$ are the corresponding quantum
mechanical amplitudes.
We assume that the wave function of the gas evolves in time
in accordance with the GRW dynamics, and that the gas is macroscopic
enough, so that there is high probability for a GRW jump to occur
during any dynamical time interval $\Delta t$ which is short in a thermodynamic scale.
When a jump occurs the wave function of the gas collapses into a state
that is localised around a certain position
$x=x_1,x_2,...,x_N$, that is, around some spatial distribution of the
gas molecules. For example, at $t_1$ the wave function collapses into some
state $\ket{\psi_1}$ which
corresponds to a Gaussian centered around $x(t_1)$. The collapsed
state then evolves
in accordance with the Schr\"odinger equation.
The high mass density of typical macroscopic
systems implies an overwhelmingly high probability for a collapse
by $t_2$ (where $t_2-t_1$ is short in thermodynamic scales) onto a Gaussian
centered around a position
$x(t_2)$, where $x(t_2)\not=x(t_1)$.
Since the GRW jumps presumably solve the measurement problem, a sequence of such jumps results in a trajectory in the system's
state space which can be described in terms of
thermodynamic magnitudes. It is therefore possible, in this case, to
determine whether or not the evolution of the system obeys the laws of
thermodynamics (\eg of entropy increase). Suppose now that we write down the
GRW equation for a given thermodynamic system, and solve it for all
possible initial states. Consider any time interval $(t_1,\,t_2)$
for which, according to the GRW prescription, a collapse of the
quantum state of the system occurs with high probability. For every possible
initial state at $t_1$ there are in general many (possibly infinitely many)
possible final states at $t_3$. For each such evolution, it is then
possible to determine whether the evolution is thermodynamic normal or
abnormal.
Albert (2000, pp. 148-162, and especially pp. 155-6) now observes
the following. First, the GRW jumps can be understood as inducing
stochastic perturbations of the quantum state of the gas.
We thus have an {\em internal perturbation mechanism}, as opposed
to the external mechanism used in classical open
system approaches. As we explained above the
wave function, say of our gas in (\ref{eq:gas}), follows a
genuinely stochastic trajectory in the system's state space.
Second, and moreover, any GRW collapse induces a set of
probability distributions, that is, transition probabilities -
given the wave function just prior to the
collapse - over the possible wave functions of the system
immediately after the collapse. So in order for a GRW collapse to put
(with high probability), our gas's wave function, on a segment of a
Schr\"odinger trajectory which is
thermodynamic normal, what is needed is that the
thermodynamic normal states (throughout the set of microstates to
which the system can collapse) overwhelmingly
outnumber\footnote{Albert (1994a,b; 2000, Chapter 7)
provides only qualitative
plausibility arguments for his approach, and so we have to follow him in using vague terms
such as the above.} the thermodynamic abnormal ones. Moreover, we need this condition
to hold in every {\em microscopic} region of the state space. If this
turns out to be correct it would mean that after a GRW jump
the wave function of the system will be (with high
probability) thermodynamic normal. This, regardless of
the history of the system, and in particular of the state of
the system immediately before the collapse.
And so, {\em each and every state} has an overwhelmingly high probability
to evolve to a thermodynamic normal state following a GRW jump.
This implies that the property of being thermodynamic normal is {\em
stable} over time, whereas that of being thermodynamic abnormal is highly
{\em unstable}. In effect, what is needed is that
the GRW probabilities for the collapse
transitions reproduce the probabilities of the
(ab)normal trajectories calculated from the standard
statistical-mechanical measure for
any given macrostate of the system. Albert puts forward the
{\em hypothesis} that as a matter of fact the GRW dynamics provides
precisely this. Call this Albert's {\em dynamical hypothesis}.
Note that Albert's hypothesis need not invoke postulates
regarding initial states and probability
distributions thereof. Rather, since the GRW dynamics is
genuinely stochastic, whether or not this hypothesis is true
depends on the set of transition probabilities generated by the GRW
collapses. In this sense, Albert's approach aims
at deriving the thermodynamic regularities from the underlying GRW
dynamics {\em only}, without recourse to initial states or
probability distributions thereof.
\subsection{Some Advantages of Albert's Approach}\label{Alb2}
{\em One Solution for Two Problems.}
In Albert's approach, the GRW solution of the measurement problem is also the solution for the problem in the foundations of statistical mechanics, namely, a dynamical justification for the use of probability. Moreover, the same system properties required for the GRW solution of the measurement problem to take place (large or massive systems) are the systems in which statistical mechanics can best recover thermodynamics.
For typical thermodynamic
systems the GRW jumps are highly probable to occur at all times, either
because such systems are typically massive enough, or because they interact with some other massive systems, such as the
interactions of the gas's molecules with the box's walls. In such cases
the GRW theory gives high probability for a collapse of the quantum wave
function, and this means that the thermodynamic system has high
probability to be at all times in
a localised state. The measurement problem is solved and the thermodynamic magnitudes are well defined. When we add Albert's dynamical hypothesis
(regarding the solutions of the GRW equations) we get the
probability distributions that are needed for statistical mechanics to work. Recall that the dynamical hypothesis still lacks proof.
But should it be proved, Albert's approach will present
a unified way in which
the GRW razor, so to speak, cuts twice: in the theory of
quantum measurement and in the foundations of statistical mechanics.
Moreover, it will have {\em two clear advantages},
both of which are related to the notion of probability
as chance, as follows.
{\em Single Origin of Chance in Physics.} First, on the GRW dynamics the
collapse of a superposition such as (\ref{eq:sup}) onto one of the terms
$\ket{\psi_i}\ten\ket{\varphi_i}$ is a purely chancy event. The jumps
invariably induce transitions from pure states to pure states, where no
crucial role is played by ignorance probabilities in mixtures.
For example, in (\ref{eq:sup}) the quantum mechanical probabilities
$\abs{\mu_i}^2$ describe irreducible and genuine chances for a
transition from the superposition (\ref{eq:sup}) to the corresponding
localised state $\ket{\psi_i}\ten\ket{\varphi_i}$.
This means that Albert's approach has a clear advantage of
parsimony.\footnote{Of course, there are other respects in which
Ockham's razor cuts against the GRW theory , \eg since the theory
postulates {\em two new} constants of nature.}
The epistemic probabilities in classical statistical mechanics are
completely reduced to the GRW quantum mechanical chances. Thus,
the classical epistemic probabilities have no essential role in physics
on this view.\footnote{This means that problems in the
interpretation of probabilities in the classical approaches, such as the
connection between probability and relative frequency distributions over
infinite ensembles, and the connection between such ensembles and the
evolution of single systems in finite times simply don't arise.}
{\em No Recourse to Probability Distributions Over Initial
Conditions.} The second advantage is this. The GRW collapse dynamics and
the probabilities for such collapses around any given spatial point at a
given time depend on the wave function of the (total) system only at {\em
that} time and
don't depend on initial conditions (or the initial wave function). In the
context of thermodynamics this means that Albert's approach need not
rely on statistical postulates regarding {\em initial} (micro)
conditions or probability distributions over them. Given the hypothesis
about the preponderance of the thermodynamic normal evolutions,
the GRW jumps have high probability to result at all times in
thermodynamic normal trajectories for a typical (macroscopic)
thermodynamic system irrespective of (micro) initial conditions.
\section{Quantum Decoherence}\label{dec}
We now consider an alternative approach to Albert's: namely,
the recovery of the thermodynamic regularities on the basis of
quantum mechanics without collapse. We shall rely on results
in decoherence theory of open (quantum) systems. In this respect our
approach belongs to the interventionist (or open systems) tradition in the
foundations of classical statistical mechanics. Let us start by briefly
describing the standard models of decoherence through the interaction with the
environment in no-collapse quantum mechanics (see \eg Zurek (1982, 1993),
Caldeira and Leggett (1983), Joos and Zeh (1985), Giulini et. al. (1996)).
This is followed by a description of results by Zurek and Paz (1994)
concerning the role of environmental decoherence in accounting for the
increase of the von Neumann entropy of quantum chaotic systems.
In the standard quantum mechanical models of decoherence
the total initial state of a macroscopic system plus environment
is usually assumed to be a product state
\begin{equation}
\ket{\psi(x_1...x_N,t)}\ten\ket{E},
\label{eq:prod}
\end{equation}
where $\ket{\psi(x_1...x_N,t)}$ is the quantum state of
the system and $\ket{E}$ is some state of the environment. This means,
in particular, that the states of the system and of the environment are
separable (\ie not quantum mechanically entangled; here to simplify
the models we assume that they are also pure states, but this is not
necessary). One of the key features in these models is that the interaction
between the system and the environment is assumed to be governed by a
Hamiltonian that commutes (approximately) with some observable of the
system. That is,
\begin{equation}
[H_{int}, \Pi]\approx 0.
\label{eq:Hint}
\end{equation}
where $H_{int}$ is the interaction Hamiltonian, and the
system observable $\Pi$ (called the {\em pointer variable})
is usually taken to be position. In this sense, the standard models of
decoherence usually assume that (approximate) position is a preferred
basis in the Hilbert space of the system.\footnote{More generally, this
basis is fixed by the dynamically conserved quatities.} In general,
the time evolved (Schr\"odinger) state can be written in the form
\begin{equation}
\ket{\Psi(t)}=\sum_i\mu_i(t)\ket{\psi_i}\ten\ket{E_i(t)},
\label{eq:dec}
\end{equation}
where the kets $\ket{\psi_i}$ are assumed to be the eigenstates of $\Pi$,
and the $\ket{E_i(t)}$ are the relative states of the environment.
The set of states $\{\ket{\psi_i}\}$ is called the {\em pointer basis}.
The result of the coupling is that the scalar products between the
environment states $\ket{E_i(t)}$ in (\ref{eq:dec}) relative
to different pointer states $\ket{\psi_i}$ decay exponentially satisfying
\begin{equation}
\braket{E_i (t+\Delta t)}{E_j (t+\Delta t)}\approx\delta_{ij}
\label{eq:Eij}
\end{equation}
after extremely short times $\Delta t$ (called {\em decoherence
times}) which are typically around
$10^{-23}{\rm sec}$.\footnote{The relaxation times of
the system are typically extremely long, in some models
of the order $10^{40}{\rm sec}$. Also, the decoherence
times of the system are
much shorter than the dynamical times even for very
weakly dissipative systems.}
The decay of the scalar products in
(\ref{eq:Eij}) is known as {\em environmental decoherence}.
Joos and Zeh (1985) derive a master equation for the
reduced state of the system assuming recoil-free scattering (\eg
large mass ratio of the decohered system over the scattered particles)
and isotropy in the distribution of the incoming particles
(photons and molecules). Under these assumptions, the solutions
of the equation exhibit exponential decay of the off-diagonal
elements. The localisation rate is proportional to
${\rm e}^{-\Lambda(x-y)^2t}$, depending in general on various factors,
such as the strength of the coupling, temperature and mass
ratios. For our purposes the following results are crucial.
First, in the standard models of decoherence, in
particular, models in which there is a pointer basis,
(\ref{eq:Eij}) and (\ref{eq:dec}) imply that
the reduced state of the decohering system approaches the diagonal form:
\begin{equation}
\rho_s(t)\approx\sum_i
\ket{\psi_i}\abs{\mu_i(t)}^2\bra{\psi_i},
\label{eq:diag}
\end{equation}
within times comparable to $\Delta t$.
Second, in these models the diagonal form (\ref{eq:diag})
of the reduced state is stable over time (\ie
the scalar products (\ref{eq:Eij}) remain vanishingly small).
Third, Zurek, Habib and Paz (1993) consider the decoherence
interaction of a harmonic oscillator with an environment in thermal
equilibrium. They show explicitly in the weak coupling limit that
the pointer states $\ket{\psi_i}$ correspond to so-called {\em coherent
states}, \ie narrowly peaked Gaussians in both position and momentum. In
their model, coherent states are the most stable states for the
system in the sense that they produce the least von Neumann and linear
entropy, so that $\rho_s(t)$ becomes maximally mixed when diagonalised
by coherent states. In this sense one can say that in the standard models
{\em decohering} systems follow quasi-classical trajectories. In order to
explain this last sentence a few more details are in place.
\subsection{Decoherence and the Von Neumann Entropy}\label{neumann}
It turns out that decoherence plays an essential role in
accounting for
the emergence of classical behaviour in quantum mechanics as well as in the
recovery of thermodynamic behaviour. In the case of {\em
chaotic} systems Zurek and Paz (1994) have shown how to recover the
classical dynamics from the underlying quantum dynamics. In their models they
show why the classical evolution cannot be recovered for closed systems,
whereas open decohering systems invariably exhibit an evolution which is
approximately classical. As we shall see (section \ref{exp}) their
results may be taken to support our proposal for recovering thermodynamics
from quantum mechanics, since it implies that the von Neumann entropy of
decohering systems increases in the course of time. Note, however,
that it is questionable whether or not the von Neumann entropy is the exact
quantum mechanical counterpart of the thermodynamic entropy.\footnote{see
Shenker (1999), Henderson (2002). We address this issue in a forthcoming
paper.}
Before
explaining this point in detail, let us describe qualitatively
how Zurek and Paz derive their results.
In the case of {\em classical} chaotic systems there are essentially
two constraints on the dynamical evolution: (i) trajectories with
initially close segments diverge exponentially; and (ii) the flow of
the probability distribution is volume preserving, by Liouville's theorem.
These two constraints together have the consequence that the accessible phase space region (containing the states which are the time evolutions of the initial states) assumes a structure which is highly striated on increasingly finer length scales, at
exponential rates.
In the case of quantum systems we have two
extreme results.
The dynamics of quantum chaotic systems on phase space is usually taken to be
described by the so-called Wigner function which yields the probability
distribution over position and momentum.\footnote{The equation of motion for
the Wigner function is given by the Moyal bracket:
$$
\{H,W\}_{mb}=-i\sin(i\hbar\{H,W\}_{pb})/\hbar,
\label{eq:mb}
$$
where $\{H,W\}_{pb}$ is the Poisson bracket describing the
classical evolution of the Wigner function $W$, and $H$ is the
Hamiltonian. This yields the evolution equation
$$
\begin{array}{rcl}
\dot W =
\{H,W\}_{pb} & + &
\sum_{n\geq 1}\frac{(-1)^n\hbar^{2n}}{2^{2n}(2n+1)!}
\partial_x^{2n+1}V(x)\partial_p^{2n+1}W + \\[2ex]
& + & 2\gamma\partial_p(pW)+d\partial^2_pW,
\end{array}
\label{eq:wig}
$$
where the first term gives the classical Liouville flow, the second (higher
derivatives term) describes the quantum mechanical corrections for the
evolution of a closed system, and the last two terms result from the
interaction with the environment field. These last terms describe,
respectively, the relaxation of the system (where $\gamma$ is the relaxation
rate), and diffusion (where $d=2\gamma m k_BT$; $m$ is the mass of
the system, $k_B$ is the Boltzmann constant, and $T$ is the temperature
of the field). The last diffusive term induces the suppression of
quantum interference (represented by the off-diagonal
elements in the reduced state $\rho_s(t)$ of the system)
in the reduced dynamics (see Zurek and Paz (1994), E. Joos
in Giulini et.al. (1996), Chapter 3, sec. 3.2.3, and references therein).}.
Note that the Wigner function cannot be straightforwardly
interpreted as a probability distribution since it takes sometimes {\em
negative} values. But for approximate measurements of position and
momentum on scales of
$\hbar^n$ it does yield (formally) a probability-like
distribution.
(1) {\em Closed systems.} If the system is completely closed
(\eg there are no decoherence interactions), the chaotic dynamics results
in exponential divergence of neighbouring trajectories.\footnote{This is
because the last two terms in the evolution equation of the Wigner
function are dropped.} But this is compensated by an exponential contraction
in the opposite directions, so that the total volume of regions in the phase
space over which the Wigner function is nonzero remains {\em constant}
throughout this evolution, in agreement with Liouville's theorem. However,
on finer length scales the contraction generates interference fringes which
the Wigner function cannot follow since it cannot be positive throughout
$\hbar^n$-sized regions of phase space. This means that the dynamics
{\em deviates} from the classical evolution. In particular, the system
doesn't follow (not even approximately) classical trajectories.
The crossover time at which the quantum corrections in
the dynamical evolution of the system become effective is approximately
\begin{equation}
t_c{\rm ln}(I/\hbar),
\label{eq:time}
\end{equation}
where $t_c$ is the divergence rate of the chaotic trajectories of
the system (\ie the time scale at which the classical dynamics
develops fine structure below $\hbar^n$-sized regions of phase space),
and $I$ is the action. The smaller the quantum corrections in the
dynamics are, the shorter this time is. In other words decoherence results in
a small delay in the divergence rate of the trajectories (see Zurek and Paz
(1994)).
(2) {\em Open systems.} When the system undergoes a decoherence
interaction with the environment there is an initial (relatively short)
time interval (shorter than (\ref{eq:time})) in which the dynamical
evolution is approximately reversible, volume preserving,
and follows the classical Liouville flow. But for times larger than
the decoherence time of the system the Wigner function delocalises
in position and decoherence is set on. This means that there is an
effective collapse of the total state (\ref{eq:dec}) onto the
corresponding mixture much before the evolution deviates from the
classical evolution.\footnote{Note that this means that the diffusion
in the evolution of the Wigner function reduces the quantum coherence
on {\em decoherence} time scales.}
As a result the Wigner function evolves towards a mixture
of localised Gaussian states on a time scale comparable to (\ref{eq:time}).
These states evolve independently of each other following
approximately the {\em classical} evolution. But since interference
terms in the reduced dynamics are washed out (in correspondence with
(\ref{eq:diag})), the total phase space volume over which the Wigner
function spreads {\em increases} monotonically. This process goes on
approximately on time scales at which equilibrium is reached.
Thus, effectively, in the presence of decoherence the
system follows quasi-classical trajectories (as it were, it
doesn't have enough time to deviate from the classical evolution).
The effect of this dynamics on the quantum mechanical
(von Neumann) entropy $-k{\rm Tr}\,\rho {\rm ln}\rho$ is this.
In case (1) of isolated systems the von Neumann entropy remains
approximately constant (and identically zero if the system
starts out in a pure state).
In case (2) Zurek, Habib and
Paz (1993) and Zurek and Paz (1994) show (in the simple
model of a decoherence interaction of a harmonic oscillator with an
environment in thermal equilibrium) that decoherence yields an increase
in the von Neumann entropy as a monotonic function of the volume in
phase space (see also Joos and Zeh 1985, pp. 235-6.)
The rate of increase of the von Neumann entropy is proportional
to the degree of mixing of $\rho_s(t)$ (depending logarithmically
on the number of eigenstates of $\rho_s(t)$). Assuming,
for instance, that the diagonal elements of $\rho_s(t)$ in
(\ref{eq:diag}) are approximately equal, the von Neumann
entropy is ${\rm ln} N$. In the case of chaotic systems the von
Neumann entropy increases (before equilibrium is reached) at a rate that
is approximately equal to the divergence rate of the
chaotic trajectories (Zurek and Paz (1994, 1995, 1999, Chapter 5); see also
Monteoliva and Paz (2000)), approaching (asymptotically) the classical
Kolmogorov-Sinai rate for entropy production (Zurek and Paz 1995). This means
that the rate at which the von Neumann entropy increases is
approximately the classical one. In the case of decohering systems which
are not chaotic, the von Neumann entropy also increases but at a much slower
rate.\footnote{The rate of increase in von Neumann entropy in the case of
chaotic systems is independent of the diffusion coefficient appearing in the
evolution of the Wigner function. In the non chaotic case the entropy
production depends only on the value of the diffusion
coefficient. See further details in Zurek and Paz (1994; 1995; 1999,
Chapter 5), and Monteoliva and Paz (2000).}
To sum up: in case (1) of isolated systems classical
thermodynamics and classical mechanics lead to predictions
that {\em disagree} with the predictions of standard
quantum mechanics. This holds as long as the system is truly
isolated from its environment so that there are no decoherence
interactions. This is correct of course only if the
von Neumann entropy does correspond to the usual notion of entropy as
it is defined in thermodynamics and classical statistical mechanics (as we
noted above this is questionable). In case (2) of open
decohering systems the classical predictions are recovered. The Wigner
function breaks
down (due to the effective collapse of the quantum wave function) to Gaussian states each of which follows approximately quasi-classical
trajectories (as given by the Poisson bracket). Because of the persistent
(effective) reduction of interference terms (in correspondence with the
diagonal form of the reduced state
$\rho_s(t)$ in (\ref{eq:diag})) the total volume of phase space
over which the Wigner function is nonzero increases monotonically (so that
Liouville's theorem no longer describes it). In these cases we obtain a monotonic
(and effectively irreversible) increase in the von Neumann entropy
at approximately the classical rates.
\section{Thermodynamics Without Collapse}\label{exp}
Our project is similar to Albert's, namely, to justify the use of
probability as it is used in classical statistical mechanics, but using
quantum mechanics as the underlying dynamics. This project is different
from the one undertaken by Zurek and Paz (1994). Their project was to
show that environmental decoherence brings about an increase in the von
Neumann entropy. We (and Albert), on the other hand, argue that
decoherence brings about an approach to equilibrium in the classical sense
of, for example, an evolution towards the most probable macrostate. The
concepts of entropy, equilibrium, etc. that we use are those appearing in
classical statistical mechanics, in either its Boltzmannian version or
the Gibbsian one. We shall be willing to use the von Neumann entropy only
to the extent that it corresponds to those classical notions, and this
correspondence is yet to be established (as we have already remarked).
What, then, is the role of the above results, concerning the von Neumann
entropy, in our argument? These results will serve as a significant {\em
support} for a hypothesis that we shall make, but the hypothesis is
reasonable on other grounds as well, and so our proposal does not {\em
depend} on the above results.
But before proceeding to undertake this project, let us make some further
remarks on the extent to which the von Neumann entropy corresponds to any
classical notion of entropy based on phase probabilities. Some problems
arise in this context.
First, decoherence by itself does not
solve the measurement problem in quantum mechanics.
The interference terms in the superposition (\ref{eq:dec}) are not
eliminated, but rather diffused into
the degrees of freedom of the environment. It is true that the effects of
interference between the different terms in (\ref{eq:dec}) are
effectively undetectable for times longer than the decoherence time
$\Delta t$ of the system, and in the pointer basis the reduced state
$\rho_s(t)$ have the form of a classical statistical mixture.
But $\rho_s(t)$ in (\ref{eq:diag}) is an {\em improper}
Mixture.
We still lack an explanation for why our experience singles out only one of the $\ket{\psi_i}$
as actually occurring on each occasion.
This means that $\rho_s(t)$ cannot be taken to
represent a probability distribution, and the diagonal elements
$\abs{\mu_i(t)}^2$ cannot be interpreted as probabilities of the
corresponding states $\ket{\psi_i}$.
Second, in the context of thermodynamics, the measurement problem
translates into a problem about the {\em meaning} of the phase space
functions and of the thermodynamic
properties of the system.
Take first the Wigner function. Due to decoherence
it behaves formally like a probability distribution whenever there
exists a stable pointer basis (of coherent states).
But even in these cases (where the Wigner function takes
only positive values, and its evolution is effectively
irreversible) the reduced state $\rho_s(t)$ still doesn't
correspond to a probability distribution. And thus the reduced
dynamics as described by the Wigner function (or by the
evolution of the reduced state $\rho_s(t)$) cannot be said
to follow quasi-classical trajectories.
Moreover, since $\rho_s(t)$ is not a
probability distribution, the von Neumann entropy too cannot be
given a Gibbsian interpretation in terms of a probability distribution.
Similarly, a Boltzmannian notion of entropy based
on dividing the diagonal elements in
$\rho_s(t)$ into sets corresponding to macrostates makes no sense. In a
Boltzmannian approach entropy is a physical relation between a given
microstate of an individual system and a given macrostate. In classical
statistical mechanics a macrostate is associated with volume in phase space,
and the entropy of the microstate of the system at a given time is the
logarithm of the standard measure of the volume which includes this
microstate at this time. Thus, it is part and parcel of the
Boltzmannian notion of entropy that the system actually {\em be} in
a given microstate (and {\em a fortiori} in a given macrostate).
This means that the von Neumann entropy (and the Wigner function)
cannot be properly understood in a Gibbsian approach, nor in a
Boltzmannian approach. In quantum mechanics the application of
such approaches requires a solution to the measurement problem.
Some of the above problems can be
solved by appealing to no-collapse interpretations of quantum
mechanics (modal, many worlds and pilot-wave theories). In such
interpretations there are extra dynamical laws
(over and above the Schr\"odinger equation) according to which
$\rho_s(t)$ in (\ref{eq:diag}) represents a genuine probability
distribution over the $\ket{\psi_i}$. The state of the system
at each time is associated with one of the states $\ket{\psi_i}$
(call them {\em effective states}) corresponding to the diagonal
elements of $\rho_s(t)$. And there are transition probabilities
between any two such effective states at different times.\footnote{The
interpretation of the reduced state as describing the single-time
probability distribution over the $\ket{\psi_i}$ is similar in the
pilot-wave, modal and many worlds theories. However, these theories
differ in their account of the multi-time (joint and transition)
probabilities.} In such interpretations decoherence is usually used in
order to explain (on the basis of the dynamics of the quantum state) why
the different terms in the time evolved superposition
(\ref{eq:dec}) effectively cease to interfere.\footnote{Note
that the pointer basis which is taken here as preferred in decomposing
$\rho_s(t)$ is fixed by the condition in (\ref{eq:Hint}).}
And then the diagonal form of $\rho_s(t)$ (as in
(\ref{eq:diag})) together with the interpretation of $\rho_s(t)$
as describing probabilities is taken to explain in such
interpretations the so-called {\em effective} collapse of
the state. As is well known any one of the above interpretations of
quantum mechanics faces its own problems (\eg in the context of
measurement theory, relativistic generalisations), and so whether or
not it may be taken as a foundation of classical statistical
mechanics will depend on how these problems will be solved.
In this paper we do not address these issues and do not advocate
choosing one of the above interpretations.
\subsection{The Proposal}\label{prop}
Consider systems which conform to the standard models of decoherence
(\eg Caldeira and Leggett (1983), Joos and Zeh (1985), Zurek, Habib and Paz
(1993)) in which there is a stable pointer basis, and in which decoherence
yields localisation of the effective states of the system.\footnote{Cases
in which decoherence does not lead to a pointer basis (of localised
states) are discussed in (HS2002b) section 5.} In the standard models the
total state of system plus environment has the form (\ref{eq:dec}), that
is
\begin{equation}
\ket{\Psi(t)}=\sum_i\mu_i(t)\ket{\psi_i}\ten\ket{E_i(t)},
\label{eq:dec2}
\end{equation}
where the effective states $\ket{\psi_i}$ diagonalizing the reduced
state of the system are {\em coherent} states. The coherent states in these models are the maximally stable states
under the time evolution (including the decoherence interaction). Figure 2 illustrates the evolution of a state during a time interval longer than the
decoherence time of the system.
The total state above evolves in accordance with the
Schr\"odinger equation.
The different terms in the superposition
(\ref{eq:dec2}) which are approximately product states of the form
\begin{equation}
\ket{\psi_i}\ten\ket{E_i(t_1)}
\label{eq:bran}
\end{equation}
evolve approximately in accordance with the Schr\"odinger free evolution, independently of each other,
because the $\ket{E_i(t_1)}$ don't re-interfere. In this sense we
obtain an effective collapse of the state (\ref{eq:dec2}).
In general, we can assume that the independent time evolution of each of the branches induces some spread
in position ($t_2$ in Figure 2). When this spread becomes larger than the
coherence length, decoherence will operate again ($t_3$ if Figure 2), and as a result the
reduced state $\rho_s(t)$ of the system will become mixed in each of the
time evolved terms (\ref{eq:bran}). But decoherence insures that
$\rho_s(t)$ will be diagonalized, again, by coherent states. And so in all
branches of the total state the new
effective states of the system at $t_3$ are coherent states
coupled to (approximately) orthogonal
$\ket{E_i(t_3)}$. The states $\ket{\psi_i}$ at $t_3$ are now centered
around some spatial points $x_{i}(t_3)$ that are, in general,
different from $x_{i}(t_1)$.
We may now assume that a single effective state at $t_1$ (\ie one
of the $\ket{\psi_i}$ at $t_1$) does not uniquely
determine a single effective state at $t_3$. That is, we assume that
the transitions between the effective states are genuinely
{\em stochastic} (the two-time correlations are not one-one).
This depends on the details of the extra dynamics of the no-collapse
theory in question: in some modal and many worlds interpretations the
dynamics is indeed genuinely stochastic.\footnote{See
Bacciagaluppi (1998) and Bacciagaluppi and Dickson (1999)
for a detailed discussion of the modal
interpretation of quantum mechanics, and in particular the extra
stochastic dynamics. In the pilot-wave
theory the velocity equation is deterministic, and so the trajectory
of the system is fixed by the initial conditions and the dynamics. In
this theory there are no genuinely stochastic transitions between
different trajectories along which the system can evolve. In the many
worlds theory the question of whether or not there are stochastic
transitions when a state of the form (\ref{eq:dec2}) branches is under
dispute.}
On this assumption the result is that the effective state of the
thermodynamic system changes in a stochastic way in the course of
decoherence. It may be convenient to think about these transitions
as if they induce random perturbations on the system's trajectory.
The crucial point is that these transitions don't in fact depend on
initial conditions (over and above those needed to secure
decoherence). In this sense they play exactly the same role played
by the GRW jumps in Albert's approach.
Suppose now that we write down the
Schr\"odinger equation for a given thermodynamic system,
and solve it for all possible initial states.
Take the time interval $(t_1, t_3)$.
For every possible initial state
at $t_1$ (which for simplicity we assume is approximately pure)
there are many possible evolutions that
branch out from it, corresponding to different relative states of the
environment (many possible final states at $t_3$). Compare all
the pairs of states, one of which is a possible initial state at $t_1$
and the other is one branching-out evolution of it at $t_3$. It is then
possible to determine, for each such pair, whether the transition from
the state at $t_1$ to the state at $t_3$ is thermodynamic
or anti-thermodynamic.
We now argue that the thermodynamic evolutions overwhelmingly
outnumber the anti thermodynamic ones.
The reason is this. The effective evolution is made of segments, each of
which is of the $t_1$-to-$t_3$ type
(in Figure 2). Due to the genuinely stochastic nature of the selection of
segments which make up the evolution, {\em if} most transitions between
effective states are thermodynamic normal, {\em then} the overall evolution will be
thermodynamic, regardless of whether or not any of the segments which make it up happens to
be thermodynamic abnormal. Recall: a trajectory is thermodynamic normal (abnormal) if the succession of states obeys (violates) the laws of thermodynamics. A prerequisite is of course that the thermodynamic magnitudes will be well defined, that is, that the measurement problem will be solved.
In order to prove the above {\em if} clause, we proceed in two
stages. First, we put forward a {\em dynamical hypothesis} that the
overwhelming majority of the above $t_1$-to-$t_3$ transitions are
thermodynamic normal. More precisely, our hypothesis says that the stochastic
transitions of the extra variables reproduce during
decoherence processes the standard measure as used in classical
statistical mechanics. If this is correct it would mean that the
decoherence interaction induce perturbations (\ie stochastic transitions
between effective states) that are enough to put the
effective wave function of the system with high probability and with
high enough rates on thermodynamic normal trajectories.
This hypothesis is a counterpart of Albert's dynamical hypothesis
(section \ref{Alb2}), and it needs of course to be
proved within a given no-collapse theory.
Second, to {\em support} our
hypothesis in the framework of quantum mechanics without collapse
we turn to the results in decoherence theory by Zurek
and Paz described in the previous section. These results demonstrate
that the process of
decoherence brings about an increase of the von Neumann entropy.
This entropy is of a Gibbsian-type. But given the proximity between the
results of applying the Gibbsian and Boltzmannian approaches in the right
circumstances,\footnote{Although they are conceptually very different, and
differ in results as well, as already emphasized by Jaynes (1965).}
it seems highly reasonable that entropy in a Boltzmannian approach
increases as well due to decoherence. Consequently, it is highly
reasonable that in the course of decoherence most of the $t_1$-to-$t_3$
type of evolutions are thermodynamic normal for {\em all} possible initial
states of the system which lead to decoherence\footnote{We discuss the
role of initial conditions in section \ref{stoch}, and in (HS2002b).}.
Note that Zurek and Paz's results support our dynamical
hypothesis only insofar as the von Neumann entropy is equivalent to or
is a counterpart of the thermodynamic entropy. This idea, however, is under
dispute (see Shenker (1999), Henderson (2002)). According to our proposal
it is possible to explain thermodynamics without recourse to the von
Neumann entropy by relying directly on our dynamical hypothesis.
\section{Probabilities and the Role of
Stochasticity}\label{stoch}
As we saw in Albert's approach the stochastic jumps of the
GRW theory (given his dynamical hypothesis) have two
important consequences: (i) the
trajectory of a system will be (with high
probability) thermodynamic normal independently of whether or not
the initial state of the system was thermodynamic normal or
abnormal (thus thermodynamics is obtained as a pure result of
the GRW dynamics); and (ii) the probabilities in classical
statistical mechanics are entirely reduced to the quantum
mechanical probabilities, as the latter are construed by the GRW
theory. This second point means that all probabilities in physics
may be construed as objective probabilities (pure chances), and
in particular it means that ignorance probabilities need
play no fundamental role in physics.
By contrast, Albert (2000, pp. 152-3) further argues that
the stochastic dynamics in no-collapse interpretations
of quantum mechanics (\eg many worlds or many minds
interpretations, or the stochastic evolution of the
extra variables in modal interpretations) will not in
general induce the right transitions required for
thermodynamic evolutions. This is because in these
interpretations the evolution of the quantum state is
given by the deterministic Schr\"odinger equation, and this
evolution also determines completely both the evolution of
the {\em probabilities} and the evolution of the set of all
physical properties of the system. One may say that in these
interpretations the set of probabilities and the set of
properties of the system provide, as it were, an envelope of
possibilities which evolves in time in a completely
deterministic fashion. Therefore, Albert concludes
that in no-collapse quantum mechanics one cannot establish
results analogous to points (i) and (ii) above, and so this
is a clear advantage of his approach.
However, the foregoing reasoning applies only to {\em isolated} quantum
systems. That is, as long as the thermodynamic system is isolated, its
behaviour is fixed entirely by the evolution of its quantum state alone.
Moreover, if by the Schr\"odinger equation, the quantum state of an
isolated system evolves along a thermodynamic abnormal trajectory,
then the system will violate the predictions of thermodynamics,
independently of whether or not other (so-called `hidden') variables of
the system undergo stochastic or deterministic
transitions.\footnote{In Bohm's theory the position of both open and
closed systems follows always deterministic trajectories. In modal
interpretations the evolution of the set of the extra variables
associated with the (global) properties of both open and closed
systems follows the Schr\"odinger evolution (\ie it is deterministic).
In the case of closed systems there are typically no stochastic
transitions also in the values of those properties. But for open
systems the dynamics of the extra variables is generally stochastic;
see Bacciagaluppi and Dickson (1999). In many worlds theories
the evolution of closed systems is always deterministic and
there is no corresponding splitting of worlds. Splitting (and in some
versions stochastic evolution) occurs only due to interactions
(typically measurement-like interactions).}
But, as argued above, in no-collapse interpretations the effects of
decoherence in the case of open systems are crucial for the
explanation of thermodynamic evolutions. In particular, since
decoherence results in stochastic transitions between the
system's effective states (at different times), the latter
states (as we explained above) evolve along trajectories which
will be in general {\em different} from the trajectory along
which the total quantum state evolves. It is correct
that the trajectory along which the total state evolves
is fixed deterministically by the initial state of the
system. But in the case of interacting systems (\ie as in
decoherence) the trajectories along which the effective states
of the system evolve are {\em not} fixed
deterministically by the initial quantum state of the system
(except in deterministic theories like Bohm's), and
they may be completely independent of it.
Let us see how these ideas can be understood in, for example, modal
interpretations. Recall our schema in Figure 2. In modal interpretations
the extra variables evolve between $t_1$ and $t_3$ in a genuinely
stochastic fashion, such that at $t_3$ one of the Gaussians is
chosen by the stochastic hidden variable. The chosen Gaussian is
referred to in modal interpretations as the {\em actual} (or effective)
state of the system. In our schema this means that the effective
transitions from one Gaussian state to the next are chancy or stochastic.
As explained in the previous section this brings about the thermodynamic
behaviour. In this way we can see how our approach yields the
consequneces (i) and (ii) of Albert's approach.
Because of the stochastic nature of the dynamics in modal
interpretations (for example), whether or
not the actual trajectory of a system is thermodynamic normal (or
abnormal) does not depend on the initial quantum state of the
system. And so the thermodynamic regularities can be explained
along the lines sketched above on a purely dynamical
basis, just as in the GRW theory. Furthermore,
when the system evolves along decohering trajectories the
probabilities in classical statistical mechanics are
entirely reduced to the quantum mechanical probabilities.
The role of stochasticity in our approach, however, is different
in one important respect from its analogue in the Albert-GRW
approach. Despite the fact that due to the interaction with the
environment the transitions between the effective states of the system
are stochastic (see Figure 2), the
evolution of the total wave function is time {\em reversible},
since it is governed by the Schr\"odinger equation. For example, the
total wave function may recohere in the future, and this will (or
will not) happen regardless of whether or not the evolution of the extra
variables is stochastic. And since the Schr\"odinger equation is
deterministic, the evolution of the total wave function depends entirely
on initial conditions (we discuss the
status of initial conditions in more detail in
(HS2002b)). For this reason the stochasticity in our approach (although
it may correspond to a genuine chancy evolution of the
extra variables) does not entail genuine thermodynamic irreversibility.
This is unlike the Albert-GRW approach. In quantum mechanics
thermodynamic (ir)reversibility is thus an outcome of the evolution
of the total wave function and does not depend on the behaviour of extra
variables. Here too these variables remain {\em hidden}.
Note that in Bohm's theory there can be no stochastic transitions or
stochastic perturbations of the trajectory of an open system
which don't completely depend on the initial conditions. Of course, our
analysis using decoherence is perfectly applicable also in this
theory. But since the theory is completely deterministic and time
reversible (both the Schr\"odinger dynamics and Bohm's velocity
equation are deterministic and time reversible), initial
conditions in the recovery of thermodynamics (such as the
initial distribution postulate $\abs{\ket{\Psi}}^2$) play
essentially the same role as in the classical approaches in
statistical mechanics.
We discuss in more detail the role of initial conditions in
both Albert's approach and in our approach in a sequel of
this paper (HS2002b).
\section{Summary}\label{sum}
Albert proposes to solve some problems in the foundations of statistical
mechanics using the GRW approach to quantum mechanics. Had this been the
only way to solve these problems, the GRW approach would have gained a
significant advantage over alternative interpretations of quantum
mechanics. For this reason it is important to learn that
no-collapse approaches to quantum mechanics can yield similar
results. In particular, the recovery of the thermodynamic
regularities in Albert's approach relies on a dynamical hypothesis
to the effect that the GRW dynamics produces the standard
(classical) probability measure over the microstates.
In our no-collapse approach a similar dynamical hypothesis is also
required with respect to effective transitions in decoherence processes.
Moreover, in both Albert's and our approach these hypotheses
are equally justified given the properties of, respectively, the GRW
dynamics and decoherence. It is significant to note that
in a no-collapse approach (unlike Albert's approach) our
hypothesis appears to be supported by the results of Zurek and Paz (1994)
(assuming - once again - that the von Neumann entropy corresponds
to the thermodynamic entropy).
As noted above, Albert's proposal
has significant points of strength: it proposes a single origin
for chance in physics, in both quantum mechanics and statistical
mechanics; this origin is dynamical;
and it entails that there is no need to postulate special initial conditions
when accounting for the low entropy in the past (although there is still
need to assume a low entropy past macrostate). Like Albert's approach, our
no-collapse proposal explains both quantum mechanical probability and
statistical mechanical probability in a unified way. Regarding the past
hypothesis, our approach requires that we assume initial conditions in which
decoherence but not recoherence takes place; this, however, may entail that
the initial conditions had been of low entropy! This surprising route to
greater unification is addressed in more detail
in the sequel of this paper (HS2002b).
It now remains to be seen how the two approaches account for some
seemingly problematic cases. For example, in the case of small and
light gases the GRW predicted rates for collapses may be extremely
low. In these cases it seems that the thermodynamic behaviour of
the gases cannot be explained by a GRW based approach. Another
example is the spin echo experiments in which the effective isolation
of the system may be problematic for a decoherence based approach.
Some of these problems are addressed in (Albert 2000). We feel,
however, that the solutions offered there are not satisfactory. In a sequel
of this paper (HS2002b) we describe these difficulties in
detail, and propose ways to solve them in Albert's GRW approach
as well as in our no-collapse approach.
\bigskip
{\bf Acknowledgement}
We thank David Albert, Guido Bacciagaluppi, Jeremy Butterfield, Itamar
Pitowsky, Katinka Ridderbos, and Professor Dieter Zeh for extremely
valuable comments. We also thank two anonymous referees for very
detailed and helpful comments.
{\bf References}
\noindent Albert, D. (1994a), `The Foundations of Quantum Mechanics and the
Approach to Thermodynamic Equilibrium', {\em British Journal for the
Philosophy of Science}, {\bf 45}, \mbox{pp. 669-677}.
\noindent Albert, D. (1994b), `The Foundations of Quantum Mechanics and the
Approach to Thermodynamic Equilibrium', {\em Erkenntnis} {\bf 41},
\mbox{pp. 191-206}.
\noindent Albert, D. (2000), {\em Time and Chance}, (Cambridge, Mass.:
Harvard University Press).
\noindent Bacciagaluppi, G. (1998), `Bohm-Bell Dynamics in the Modal
Interpretation', in Dieks and Vermaas (eds.) (1998),
\mbox{177-211}.
\noindent Bacciagaluppi, G. and Dickson, M. (1999), `Dynamics for
Modal interpretations', {\em Foundations of Physics} {\bf 29},
\mbox{pp. 1165-1201}
\noindent Bell, J.S (1987), `Are There Quantum Jumps',
in: J.S. Bell,
{\em Speakable And Unspeakable in Quantum Mechanics}
(Cambridge: Cambridge University \linebreak
Press, 1987), \mbox{pp. 201-212}.
\noindent Brown, H. and Uffink J. (2002), `The Origins of
Time-Asymmetry in Thermodynamics: the Minus First Law',
{\em Studies in History and Philosophy of Modern Physics},
forthcoming.
\noindent Bricmont, J., Durr, D., Galavotti, M.C., Ghirardi, G.,
Petruccione, F. and Zanghi, N. (2001), {\em Chance in Physics:
Foundations and Perspectives} \linebreak (Springer).
\noindent Bub, J. (1997), {\em Interpreting the Quantum World}
(Cambridge: Cambridge University Press).
\noindent Caldeira, A. O. and Leggett, A. J. (1983), `Path Integral Approach
to Quantum Brownian Motion', {\em Physica} {\bf 121 A},
\mbox{pp. 587-616}.
\noindent Cushing, J., Goldstein, S. and Fine, A. (eds.) (1996), {\em Bohmian
Mechanics and Quantum Theory: An Appraisal} (Dordrecht: Kluwer).
\noindent DeWitt, B.S. and Graham, R.N. (eds.) (1973), {\em The
Many-Worlds Interpretation of Quantum Mechanics} (Princeton: Princeton
University Press).
\noindent Dieks, D. and Vermaas, P. (eds.) (1998), {\em The Modal
Interpretation of Quantum Mechanics} (The
Netherlands: Kluwer).
\noindent Earman, J. and Norton, J. (1998), `Exorcist XIV: the Wrath of
Maxwell's Demon' {\em Studies in the History and Philosophy of Modern
Physics}, Part I {\bf 29}, \mbox{435-471}; Part II {\bf 30}, \mbox{1-40}.
\noindent Earman, J. and Redei, M. (1996), `Why Ergodic Theory
Does Not Explain
the Success of Equilibrium Statistical Mechanics', {\em British Journal
for the Philosophy of Science} {\bf 47}, \mbox{pp. 63-78}.
\noindent Ghirardi, G.C. (2000), `Beyond
Conventional Quantum Mechanics', in
J. Ellis and D. Amati (eds.), {\em Quantum Reflections}
(Cambridge: Cambridge University Press), \mbox{pp. 79-116}.
\noindent Ghirardi, G.C., Rimini, A.\ and Weber, T.\ (1986), `Unified
Dynamics for Microscopic and Macroscopic Systems', {\em Physical Review}
{\bf D 34}, \mbox{pp. 470-479}.
\noindent Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.
and Zeh, H. (1996), {\em Decoherence and the Appearance of a Classical
World in Quantum Theory} (Berlin: Springer).
\noindent Guttmann, Y. (1999), {\em The Concept of Probability in Statistical
Physics} (Cambridge: Cambridge University Press).
\noindent Healey, R. and Hellman, G. (eds.) (1998),
{\em Quantum Measurement: Beyond Paradox}.
{\em Minnesota Studies in Philosophy of Science}, vol. {\bf 17}
(Minnesota University Press).
\noindent Hemmo, M. and Shanker, O. R. (2001), `Can We Explain
Thermodynamics by Quantum Decoherence?', {\em Studies in History and
Philosophy of Modern Physics}, {\bf 32}(4), \mbox{pp.555-568}.
\noindent Hemmo, M. and Shanker, O. R. (2002b), `Quantum Decoherence and
the Approach to Equilibrium (Part II)', {\em Philosophy of Science},
forthcoming.
\noindent Henderson, L. (2002), `The von Neumann Entropy: A Reply to
Shenker', {\em British Journal for the Philosophy of Science},
forthcoming.
\noindent Jaynes, E.T. (1965), `Gibbs vs. Boltzmann
Entropies', {\em American Journal of Physics} {\bf 33}, \mbox{p. 391}.
\noindent Joos, E. and Zeh, D. (1985), `The Emergence of Classical Properties
Through Interaction with the Environment, {\em Zeitschrift f\"ur Physik}
({\em Z. Phys}) {\bf B} (condensed Matter) {\bf 59}, \mbox{pp. 223-43}.
\noindent Monteoliva, D. and Paz J. P. (2000), `Decoherence and the Rate of
Entropy Production in Chaotic Quantum Systems', {\em Physical Review
Letters} {\bf 85} (16), \mbox{pp.3373-3376}.
\noindent Paz, J. P. and Zurek, W. (1999), `Environment-Induced Decoherence
and the Transition from Quantum to Classical', Lectures given at the
72nd Les Houches Summer School on `Coherent Matter Waves', July-August
1999: quant-ph/0010011.
\noindent Pearle, P. (1997), `Tails and Tales and Stuff and Nonsense', in
R. S. Cohen, M. Horne, and J. Stachel (eds.), {\em
Quantum Mechanical Studies for Abner Shimony: Experimental Metaphysics}
(Dordrecht: Reidel, Boston Studies in the Philosophy of Science),
\mbox{pp. 143-56}.
\noindent Pearle, P., Ghirardi, G., and Grassi, R. (1990), `Relativistic
Dynamical Reduction Models: General Framework and Examples',
{\em Foundations of Physics}, {\bf 20}(11), \mbox{pp. 1271-1316}.
\noindent Ridderbos, K. (2002), `The Coarse Graining Approach to Statistical
Mechanics: How Blissful is Our Ignorance?', {\em Studies in
the History and Philosophy of Modern Physics}, {\bf 33}(1),
forthcoming.
\noindent Shenker, O.R. (1999), `Is $-k\,{\rm Tr}(\rho{\rm ln}\rho)$
the entropy in quantum mechanics?', {\em British Journal for the
Philosophy of Science} {\bf 50}, \mbox{pp. 33-48}.
\noindent Sklar, L. (1993), {\em Physics and Chance} (Cambridge:
Cambridge University Press).
\noindent Uffink, J. (2001),
`Bluff Your Way in the Second Law of Thermodynamics',
{\em Studies in History and Philosophy of Modern Physics}
{\bf 32}(3), forthcoming.
\noindent Walker, G.H. and Ford, J. (1969), `Amplitude Instability and
Ergodic Behaviour for Conservative Nonlinear Oscillator Systems',
{\em Physical Review} {\bf 188}, \mbox{pp. 416-32}.
\noindent Wallace, D. (2001), `Implications of Quantum Theory in the
Foundations of Statistical Mechanics' {\em Pittsburgh PhilSci
Archive} http://philsci-archive.pitt.\linebreak edu/.
\noindent Zeh, D. (1992), {\em The Physical Basis of the Direction of Time},
2nd edition (Berlin, New York: Springer-Verlag).
\noindent Zurek, W.H. (1982), `Environment-Induced Superselection
Rules', {\em Physical Review} {\bf D 26}, \mbox{1862-1880}.
\noindent Zurek, W.H. (1993), `Preferred States, Predictability, Classicality,
and the Environment-Induced Decoherence', {\em
Progress in Theoretical Physics} {\bf 89}, \linebreak
\mbox{281-312}.
\noindent Zurek, W.H., Habib, S. and Paz, J.P. (1993), `Coherent
States via Decoherence', {\em Physical Review Letters} {\bf 70},
\mbox{pp. 1187-1190}.
\noindent Zurek, W.H. and Paz, J.P. (1994), `Decoherence, Chaos, and
the Second Law', {\em Physical Review Letters} {\bf 72} (16),
\mbox{pp. 2508-11}.
\noindent Zurek, W.H. and Paz, J.P. (1995), `Quantum Chaos: a Decoherent
Definition', {\em Physica} {\bf D 83}, \mbox{pp. 300-8}.
\end{document}