# Prediction and Retrodiction in Statistical Mechanics from the Principle of
Maximum Caliber

Ignacio Tapia<EMAIL_ADDRESS>Gonzalo Gutiérrez
<EMAIL_ADDRESS>Sergio Davis<EMAIL_ADDRESS>
###### Abstract

A statistical, path-dependent framework to describe time-dependent macroscopic
theories using the Principle of Maximum Caliber is presented. By means of this
procedure, it is possible to infer predictive non-equilibrium statistical
mechanical models from a variational principle, provided that the adequate
time-dependent constraints and the state of the system at some specific times
are given. The approach is exemplified by obtaining the description of a time-
dependent Brownian particle from kinetic restrictions. We relate the
predictive nature of a model to the structure of the prior distribution that
represents the state of knowledge about the system before the dynamical
constraints are considered. Non-predictive models are shown to be possible in
the presented framework and as an example, retrodictive dynamics are obtained
from the same kinetic constraints.

###### keywords:

Non-Equilibrium Statistical Mechanics , Principle of Maximum Caliber ,
Brownian Motion , Stochastic Processes , Time-dependent Diffusion ,
Retrodiction

††journal: arXiv

[dfc]organization=Departamento de Física, Facultad de Ciencias, Universidad de
Chile,addressline=Las Palmeras 3425, Ñuñoa, city=Santiago, postcode=, state=,
country=Chile

[p2mc]organization=Centro de Investigación en la Intersección entre Física de
Plasmas, Materia y Complejidad (P2mc), Comisión Chilena de Energía
Nuclear,addressline=Casilla 188-D, city=Santiago, postcode=, state=,
country=Chile

##  1 Introduction

The aim of Statistical Mechanics, as the Encyclopedia Britannica states, is
“to predict and explain the measurable properties of macroscopic systems on
the basis of the microscopic constituents of those systems”. Due to the large
number of particles (“microscopic constituents”) involved, one needs to make
use of probability theory and statistics to _estimate_ the macroscopic
quantities in the form of expectations. Equilibrium Statistical Mechanics, in
particular, follows this program, and was put in firm basis thanks to the
works of Boltzmann, Gibbs and others at the beginning of the last century [1].
Their hypothesis, useful to characterize the macroscopic state of systems in
equilibrium with constraints, can be summarized in what is known as the
Principle of Maximum Entropy (MaxEnt). This variational principle proposed by
Jaynes [2] as a generalization of Boltzmann and Gibbs ideas, makes it possible
to estimate the macroscopic behavior of a system from specific given
information.

Non-Equilibrium Statistical Mechanics (NESM), on the other hand, deals with
macroscopic states that depend on time [3]. Although there is no a general,
accepted theory so far, the usual approach for the study of these regimes
consist of _predicting_ how a probability distribution evolves from a known
initial condition. Several schemes have been put forward to study the time
evolution of probability density functions. This kind of approach treats time
as a label unrelated to the microscopic states, which are considered as random
variables. The Principle of Maximum Caliber (MaxCal), proposed also by Jaynes
[4], has been shown to be a generalization of MaxEnt for dynamical random
variables that can be used to estimate the time-dependent behavior of
macroscopic quantities from known constraints. This principle has emerged as a
strong candidate for a description of NESM. In fact, in the last ten years the
number of practitioners has augmented, and several recent excellent reviews
exist, mainly by the group of Ken Dill and collaborators [5, 6]. There, it can
be seen that MaxCal has been used to describe different previously reported
models in a more transparent way [7, 8, 9, 10], as well as to other, new
problems, even beyond physics [11, 12, 13].

In this paper, we show explicitly how the Principle of Maximum Caliber can be
used to describe time-dependent, macroscopic systems. We focus on providing a
description for predictive NESM theories, but we also show that the path-
dependent formalism allows for the characterization of more general, non-
predictive macroscopic theories. We do this by studying the case of
retrodiction, a type of estimation where the macroscopic state of the system
is inferred from information about its future. Although less common than
prediction, retrodiction has been studied both in statistical physics [14, 15,
16] and in quantum mechanics [17, 18, 19].

##  2 Time-dependent probabilities

Among the macroscopic theories to study time-dependent probability
distributions, perhaps the best known are the Kinetic Theory of gases and the
formalism of Stochastic Processes. These theories can be used to describe
different macroscopic regimes for time-dependent systems. A system is
completely described when the time-dependent probability distribution
$\rho(x|t)$ is specified. Once the distribution is found for every time $t$,
it can be used to estimate macroscopic properties of the system in the form of
expectations

$\left\langle f(x)\right\rangle_{t}=\int f(x)\rho(x|t)\,dx\,\text{.}$ (1)

A specific theory must provide an expression for the macroscopic properties
$f(x)$ as a function of the microscopic degrees of freedom, but it also must
propose a model for the temporal behavior of the distribution $\rho(x|t)$. In
general, theories in NESM are predictive. This means that they can be used to
estimate the time-dependent distribution $\rho(x|t)$ from an initial condition
$\rho(x|t=0)=\rho_{0}(x)$. The initial condition is arbitrary. It cannot be
obtained from the model and only when it is specified (from experimental
observation or physical suppositions) the theory leads to a complete
description of the system. Therefore the model and the initial condition can
be thought of as different kinds of knowledge. It is important to distinguish
between these two aspects that are common to every NESM theory. This is
because the model is a consequence of the theory and is related to general
physical laws, while the initial condition is intrinsic to each system and has
to be known or assumed from observation.

For evolution equations that are linear on the probability density function,
such as the ones derived from a continuity equation, the dynamics can be
condensed into a master equation

$\rho(x|t)=\int P(x;x^{\prime}|t)\rho(x^{\prime}|0)\,dx^{\prime}\,\text{,}$
(2)

that is, the probability distribution $\rho$ that represents the state of the
system at time $t$ can be written as a linear functional of the initial
conditions. The kernel $P(x;x^{\prime}|\Delta t)$ is known as the transition
probability, and is the solution for the system with localized initial
conditions $P(x;x^{\prime}|0)=\delta(x-x^{\prime})$. The transition
probability, once found, can be used to construct the general solution for the
dynamics. This means the transition probability characterizes the model.

For some theories, the transition probability can be written as a path
integral [20] that can be related to the differential equation of the system
by the Feynman-Kac formula [21]. Path integrals or functional integrals
provide a mathematical tool to characterize time-dependent macroscopic states.
This formulation shows a route for an ensemble perspective to describe NESM
where the temporal dependence of the system is described in the microscopic
degrees of freedom instead of described as a macroscopic parameter.

##  3 Path-dependent probabilities

The macroscopic behavior of a system follows from the description of its
microscopic constituents. In the case of mechanical systems, this description
is usually written in a differential language, providing an evolution equation
for the microscopic degrees of freedom that acts as a constraint to the
dynamics of the macroscopic properties. The time-dependent statistical
description shown in the previous section does not provide an explicit
representation of these microscopic dynamics, because the microstates do not
depend explicitly on time. The lack of a dynamical representation of the
microscopic random variables obscures the identification of the constraints
that induce a given macroscopic behavior. Also, the representation of some
macroscopic properties is different for different time-dependent theories, as
for example the flux (or current density), for which a proper microscopic
representation would involve derivatives of the microstates.

In order to construct a dynamical model from constraints, we will consider a
statistical description, but instead of having a time-dependent probability
distribution characterizing the system, we will consider time as a parameter
of the microscopic states. We will consider the random variable of this
framework to be a function of time $\tilde{x}$ that for each time $t$ takes a
definite value $\tilde{x}(t)=x$. A time-dependent random variable of this kind
will be called a path.

The dynamics of the system will be characterized by assigning a weight
$\varrho[\tilde{x}]$, in fact a path probability density, to each possible
path $\tilde{x}$. Time-dependent expectations of observables $f[\tilde{x}]$
can be computed as path integrals in the form

$\left\langle
f(t)\right\rangle=\int\varrho[\tilde{x}]f[\tilde{x}]\,\mathcal{D}\tilde{x}\,\text{.}$
(3)

The observable inside the expectation can depend on the path evaluated at some
specific time $f[\tilde{x}]=g(\tilde{x}(t))$, but in general it is a
functional of the path.

The advantages of the path integral formalism in comparison with the time-
dependent statistics perspective relies on the direct relation between the
microscopic and macroscopic variables of the system. In quantum mechanics,
this perspective avoids the operator representation of observables, as they
can be expressed directly in the path representation as a function of phase
space variables. In NESM, the same can be done to compute expectations of
quantities that do not appear in the usual time-dependent description.
Moreover, the characterization of the dynamics using a single non-dynamical
representation $\varrho$ enables the construction of the different models from
a variational principle.

##  4 Marginalization and time-dependent macroscopic description

The path-dependent probability density defined in the previous section assigns
a weight to every possible path $\tilde{x}$. Using this distribution, we can
construct a time-dependent probability distribution that corresponds to the
representation of the macroscopic state used in the literature. This can be
done by the process of marginalization and can be written using the
expectation of a delta function

$\rho(x|t)=\int\varrho[\tilde{x}]\delta(\tilde{x}(t)-x)\,\mathcal{D}\tilde{x}\,\text{.}$
(4)

Equation (4) can be read as follows: the probability distribution that
describes the behavior of measuring $x$ at time $t$ is equal to the sum of the
probability of every path $\tilde{x}$ that is compatible with that
measurement, i.e., every path that satisfies $\tilde{x}(t)=x$.

The normalization of the time-dependent probability density $\rho$ is
inherited from the normalization of the path-dependent probability density
$\varrho$. Also, from the definition it can be proved [10] that it satisfies
the continuity equation

$\frac{\partial\rho}{\partial t}=-\frac{\partial J}{\partial x}\,\text{,}$ (5)

where the flux $J$ can be expressed also as an expectation over paths

$J(x|t)=\int\varrho[\tilde{x}]\dot{\tilde{x}}(t)\delta(\tilde{x}(t)-x)\,\mathcal{D}\tilde{x}\,\text{.}$
(6)

In the time-dependent description, to provide an explicit representation for
the flux, it is necessary to specify a theory to model the temporal behavior
of the system. The expression found here is universal to every Non-Equilibrium
theory.

In the path-dependent description, both the time-dependent density and the
flux can be expressed as functions of the path distribution $\varrho$. This
implies that the path distribution contains not only the information about the
physical laws that govern the behavior of the system, but it also contains the
information about the initial conditions of the system.

##  5 Macroscopic Constraints and the Principle of Maximum Caliber

We will consider the macroscopic constraints to be time-dependent expectations
of the form

$\left\langle f_{n}[\tilde{x}](t)\right\rangle=\bar{f}_{n}(t)\,\text{,}$ (7)

where the index $n$ distinguishes between different constraints. We can find
several path-dependent distributions $\varrho[\tilde{x}]$ that match these
constraints. The Principle of Maximum Caliber states that, from all the
possible choices $\varrho$ that are compatible with the restrictions, the most
unbiased is the one that maximizes the Caliber $C$ given by

$C[\varrho]=-\int\varrho[\tilde{x}]\ln{\frac{\varrho[\tilde{x}]}{\pi[\tilde{x}]}}\,\mathcal{D}\tilde{x}\,\text{.}$
(8)

The Caliber is a generalization of the Entropy for situations where the random
variable is a path. The definition (8) contemplates a prior distribution
$\pi[\tilde{x}]$ that encapsulates our knowledge about the system without
considering constraints. The maximization of the Caliber updates our
information from the prior distribution $\pi[\tilde{x}]$ to the posterior
distribution $\varrho[\tilde{x}]$ in order to include the constraints.

We can include the normalization condition and the macroscopic constraints (7)
with the help of Lagrange multipliers $\lambda$ and $\beta_{n}(t)$. The
Lagrange multipliers $\beta_{n}(t)$ associated with time-dependent
restrictions also depend on time, and must be included as an integral. The
auxiliary function to maximize corresponds to

$\bar{C}[\varrho]=C[\varrho]-\lambda\left\langle
1\right\rangle-\sum_{n}\int\beta_{n}(t)\left\langle
f_{n}[\tilde{x}](t)\right\rangle\,dt\,\text{.}$ (9)

The solution to the Euler-Lagrange equation associated with this functional is
given by

$\varrho[\tilde{x}]=\frac{\pi[\tilde{x}]}{Z}\exp{\left(-\sum_{n}\int\beta_{n}(t)f_{n}[\tilde{x}](t)\,dt\right)}\,\text{,}$
(10)

with the partition function $Z$ guaranteeing normalization expressed as

$Z=\int\pi[\tilde{x}]\exp{\left(-\sum_{n}\int\beta_{n}(t)f_{n}[\tilde{x}](t)\,dt\right)}\,\mathcal{D}\tilde{x}\,\text{,}$
(11)

and the Lagrange multipliers $\beta_{n}$ related formally to the constraints
as

$\bar{f}_{n}(t)=-\frac{\delta}{\delta\beta_{n}(t)}\ln{Z}\,\text{.}$ (12)

With the help of the Principle of Maximum Caliber, the choice of constraints
determines an unique manner in how the prior distribution is updated. This is
analogous to how a specific time-dependent theory updates an initial condition
into a time-dependent distribution that can be used to predict macroscopic
properties. In this context, the Maximum Caliber Principle directly relates
the macroscopic information about a physical regime to the dynamical model
that objectively best describes its behavior.

We will think of the prior distribution as a different kind of knowledge about
the system than the constraints: the prior distribution will be used to
include information about a specific system; the constraints will account for
the physical laws that produces a specific time-dependent model. For this
reason, in the context of NESM, we will assume a prior distribution equal to
the initial conditions of the time-dependent distribution
$\pi[\tilde{x}]=\rho_{0}(\tilde{x}(0))$.

##  6 Prior Distribution and Path Integration

The specific choice of a procedure to solve the path integrals involved in the
theory represent another constraint to the system. This choice will provide
the system with a specific microscopic description of the structure of paths.
Here we will treat path integration as it was first introduced into Quantum
Mechanics by Feynman [22]. That is, we will think of paths $\tilde{x}$ as
continuous but non-differentiable functions in order to have independence
between degrees of freedom associated with different times. If we take the
paths to be defined in the interval $[0,T]$, this independence can be written
formally as $\mathcal{D}\tilde{x}=\prod_{t=0}^{T}d\tilde{x}(t)$.

Usually path integrals are computed taking the end values of the paths fixed.
In the framework proposed here, paths are as general as possible to account
for a statistical description. We will not consider fixed end values in the
definition of the path integration, but the fixed points will appear in the
computation of expectations, due to the nature of the estimations performed.

The predictive nature ensured by the structure of the prior distribution can
lead to a description where every expectation has a fixed initial value. Given
that the prior distribution $\pi[\tilde{x}]=\rho_{0}(\tilde{x}(0))$ depends
only on the initial values of paths, we can introduce a delta function
$\delta(x^{\prime}-\tilde{x}(0))$ to write it as

$\pi[\tilde{x}]=\int\rho_{0}(x^{\prime})\delta(x^{\prime}-\tilde{x}(0))\,dx^{\prime}\,\text{.}$
(13)

Noting that $x^{\prime}$ is an introduced parameter, independent to the paths,
it does not participate in any expectation. This means that every expectation
can be obtained from the path distribution with a localized prior distribution
$\pi[\tilde{x}]=\delta(x^{\prime}-\tilde{x}(0))$. To include arbitrary initial
conditions, the expectations obtained have to be convolved with this initial
condition according to the marginalization rule [23],

$\left\langle f[\tilde{x}]\right\rangle=\int\left\langle
f[\tilde{x}]\right\rangle_{\tilde{x}(0)=x^{\prime}}\rho_{0}(x^{\prime})\,dx^{\prime}\,\text{,}$
(14)

where $\left\langle\ldots\right\rangle_{\tilde{x}(0)=x^{\prime}}$ denotes an
expectation taken over paths with fixed initial values
$\tilde{x}(0)=x^{\prime}$. In particular, the time-dependent distribution can
be written as

$\rho(x|t)=\left\langle\delta(\tilde{x}(t)-x)\right\rangle=\int\left\langle\delta(\tilde{x}(t)-x)\right\rangle_{\tilde{x}(0)=x^{\prime}}\rho_{0}(x^{\prime})\,dx^{\prime}\,\text{.}$
(15)

Equation (15) is the same as equation (2), implying that the transition
probability of the model can be written as a path integral with two fixed
points, corresponding to

$\displaystyle P(x;x^{\prime}|t)$ $\displaystyle=$
$\displaystyle\left\langle\delta(\tilde{x}(t)-x)\right\rangle_{\tilde{x}(0)=x^{\prime}}$
(16) $\displaystyle=$
$\displaystyle\frac{1}{Z}\int_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(t)=x}\exp{\left(-\sum_{n}\int_{0}^{T}\beta_{n}(t)f_{n}[\tilde{x}](t)\right)}\,\mathcal{D}\tilde{x}\,\text{,}$
(17)

where the fixed values were explicitly included in the path integral.

The predictive nature of the models constructed relies on the structure of the
prior distribution. We say it is predictive, because it depends only on the
initial values of paths, thus the estimations performed concern the future of
the system. Nevertheless, we will show that other structures can be chosen for
the prior distribution to obtain non-predictive theories.

We have shown here how to include the time-dependent initial conditions in a
path-dependent statistical description of NESM. The information that accounts
for the temporal behavior of the system, necessary to complete a description
and to perform predictions, must be included in the specific choice of
constraints.

##  7 Example: Free Particle in a heat bath

We will consider a one-dimensional free particle of mass $m$ in the presence
of a heat bath of temperature $\tau(t)$. The heat bath will be introduced in
the macroscopic description of the particle as a constraint on the kinetic
energy

$\frac{m}{2}\left\langle\dot{\tilde{x}}^{2}(t)\right\rangle=f(t)\,\text{.}$
(18)

If we maximize the Caliber considering this expectation as a restriction, we
obtain the path probability distribution given by

$\varrho[\tilde{x}]=\frac{\pi[\tilde{x}]}{Z[\beta]}\exp{\left(-\frac{m}{2}\int\beta(t)\dot{\tilde{x}}^{2}(t)\,dt\right)}\,\text{.}$
(19)

The partition function can be written as

$Z[\beta]=\int\exp{\left(-\frac{m}{2}\int\beta(t)\dot{\tilde{x}}^{2}(t)\,dt\right)}\pi[\tilde{x}]\,\mathcal{D}\tilde{x}\,\text{,}$
(20)

and the expectation value of the kinetic energy relates formally to the
partition function by

$\frac{m}{2}\left\langle\dot{\tilde{x}}^{2}(t)\right\rangle=-\frac{\delta}{\delta\beta(t)}\ln{Z[\beta]}\,\text{.}$
(21)

The most probable paths satisfy the differential equation

$\frac{d}{dt}\left(\beta(t)\dot{\tilde{x}}(t)\right)=0\,\text{,}$ (22)

which has an infinite amount of solutions without providing boundary
conditions.

The Lagrange multiplier $\beta(t)$ will not be adjusted using the expectation
restricted, as we will see that this expectation diverges. However, we will
see that this parameter can be related to the temperature of the heat bath
$\tau$ through the dynamics of the time-dependent marginalized distribution.

If we take a localized predictive prior
$\pi[\tilde{x}]=\delta(\tilde{x}(0)-x^{\prime})$, then the posterior
distribution $\varrho$ obtained in (19) corresponds to a Gaussian distribution
on the velocity of the particle. The mean velocity is zero, and the values of
the velocity at different times are uncorrelated, satisfying the relation

$\left\langle\dot{\tilde{x}}(t)\dot{\tilde{x}}(t^{\prime})\right\rangle_{\tilde{x}(0)=x^{\prime}}=\frac{\delta(t-t^{\prime})}{m\beta(t)}\,\text{.}$
(23)

With localized initial conditions $\tilde{x}(0)=x^{\prime}$, the most probable
path is the constant solution $\tilde{x}(t)=x^{\prime}$. It satisfies the
differential equation (22) with the least possible value for the kinetic
energy.

The path integrals are solved by taking a time-slice approach, as shown in the
majority of textbooks (see [20], chapter 2). By taking partitions
$\left\\{t_{n}=nT/N\right\\}_{0<=n<=N}$ of the interval $[0,T]$ we can express
the path integral as the limit of a $N$-dimensional vector integral. The
result obtained corresponds to

$Z[\beta]=\lim_{N\to\infty}\sqrt{\prod_{n=1}^{N}\frac{\pi
T}{mN\beta(t_{n})}}\,\text{.}$ (24)

In the continuum limit $N\rightarrow\infty$, the partition function goes to
zero. This implies that the path distribution diverges for every path. As
neither the partition function nor the path distribution are measurable
quantities, we will not ask for these aspects of the theory to converge.

The expectation value of the kinetic energy can be computed directly from the
discrete expression (24), using the relation (21) to obtain

$\frac{m}{2}\left\langle\dot{\tilde{x}}^{2}(t)\right\rangle_{\tilde{x}(0)=x^{\prime}}=\lim_{N\to\infty}\frac{N}{T}\frac{1}{2\beta(t)}\,\text{.}$
(25)

This quantity again is not convergent in the continuum limit. Nevertheless,
this divergence corresponds to a representation of the delta function
$\delta(0)=\lim_{N\to\infty}N/T$, and is consistent with the correlations of
velocities from equation (23).

Although the Lagrange multiplier $\beta(t)$ cannot be estimated from this
relation, as there is a divergence, it can be used to justify that a constant
temperature in the heat bath $\tau(t)=\tau_{0}$ corresponds to a constant
value for $\beta(t)=\beta_{0}$. We will still consider its temporal dependence
to provide a more general approach.

The time-dependent marginalization, for localized initial conditions,
corresponds to the transition probability for the predictive system. It was
computed using the same discrete scheme as with the partition function and was
found to be finite in the continuum limit, corresponding to

$P(x;x^{\prime}|t)=\frac{1}{\sqrt{4\pi\int_{0}^{t}dt^{\prime}/m\beta(t^{\prime})}}\exp{\left(-\frac{\left(x-x^{\prime}\right)^{2}}{2\int_{0}^{t}dt^{\prime}/m\beta(t^{\prime})}\right)}\,\text{.}$
(26)

This is a Gaussian distribution, with constant mean (the same as the most
probable path) and an increasing variance, given by

$\sigma^{2}(t)=\frac{1}{m}\int_{0}^{t}\frac{dt^{\prime}}{\beta(t^{\prime})}\,\text{.}$
(27)

The transition probability (26) is the kernel for a time-dependent diffusive
system. The system can be interpreted as a time-dependent Brownian motion
model, related to the Gaussian processes used in statistical learning. The
parameter $\beta(t)$ can be adjusted from the evolution of the width of an
arbitrary initial condition. The width always grows with time for positive
$\beta$ and it depends only on information about the past, as the sum does
only involve values of $\beta$ evaluated at times smaller than $t$.

The Lagrange multiplier $\beta$ is related to the diffusion coefficient $D$ of
the system by the expression $\beta=1/mD$. In this way, the Lagrange
multiplier is linked to the temperature through Einstein’s relation.

The general solution $\rho(x|t)$ with arbitrary initial conditions
$\rho(x|t=0)=\rho_{0}(x)$ can be written as a convolution given by equation
(15) and satisfies the time-dependent diffusion equation

$\frac{\partial\rho}{\partial
t}=\frac{1}{m\beta(t)}\frac{\partial^{2}\rho}{\partial x^{2}}\,\text{.}$ (28)

##  8 Retrodiction in Macroscopic Systems

As we mentioned, the structure of the prior leads to different types of
estimations. Prediction is the case when the estimations are made from
information about the past. Explicitly, in this case, the prior distribution
depends only on the paths evaluated at an initial time $\tilde{x}(0)$. If we
choose the prior distribution to depend only on the values of paths at final
times $\tilde{x}(T)$ instead, we obtain a backwards process identical to the
predictive process, but with reversed time. This backward process does not
describe a physical situation, regardless it is consistent with the prior
information assumed.

The procedure to make inference about the past with information about the
present or the future is called retrodiction and has been studied in the last
decades. It is not obtained by simply taking a final condition instead of an
initial condition for the time-dependent distribution $\rho(x|t)$. This is
because the causal assumption that the system evolves from a definite known
initial condition is not only a consideration about our knowledge, but also a
statement about the behavior of physical systems: macroscopic time-dependent
systems evolve from the past to the future, regardless of our knowledge. If we
include information about the final state of the system, we have to also
include information about an initial state.

Choosing a prior distribution that depends on the initial and final values of
paths $\pi[\tilde{x}]=\pi_{0}(\tilde{x}(0),\tilde{x}(T))$, we obtain
retrodictive models. Taking the expectations over paths with fixed boundary
values $\tilde{x}(0)=x^{\prime}$ and $\tilde{x}(T)=x^{\prime\prime}$, we can
write every expectation as a convolution

$\left\langle f[\tilde{x}]\right\rangle=\int\int\left\langle
f[\tilde{x}]\right\rangle_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(T)=x^{\prime\prime}}\pi_{0}(x^{\prime},x^{\prime\prime})\,dx^{\prime}\,dx^{\prime\prime}\,\text{,}$
(29)

where the expectation
$\left\langle\ldots\right\rangle_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(T)=x^{\prime\prime}}$
has to be taken considering a localized prior
$\pi[\tilde{x}]=\delta(\tilde{x}(0)-x^{\prime})\delta(\tilde{x}(T)-x^{\prime\prime})$.
The time-dependent marginalization
$\left\langle\delta(\tilde{x}(t)-x)\right\rangle_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(T)=x^{\prime\prime}}$
acts again as a transition probability, but this time the convolution has to
be taken over the joint probability $\pi_{0}(x^{\prime},x^{\prime\prime})$. In
the case where our knowledge about initial and final state are independent,
this joint distribution takes the form
$\pi_{0}(x^{\prime},x^{\prime\prime})=\rho(x^{\prime}|0)\rho(x^{\prime\prime}|T)$.

###  8.1 Retrodictive estimations for the free particle example

We showed that if we take as a constraint the expectation value of the kinetic
energy, we obtain the path-dependent distribution given by equation (19).
Taking a retrodictive localized prior distribution
$\pi[\tilde{x}]=\delta(\tilde{x}(0)-x^{\prime})\delta(\tilde{x}(T)-x^{\prime\prime})$,
we computed again the same quantities that we computed previously for a
predictive system.

The partition function can be expressed as a limit given by

$Z[\beta]=\lim_{N\to\infty}\sqrt{\prod_{n=1}^{N-1}\frac{\pi
T}{mN\beta(t_{n})}}\exp{\left(-\frac{\left(x^{\prime\prime}-x^{\prime}\right)^{2}}{2\sum_{n=1}^{N}\Delta
t/m\beta(t_{n})}\right)}\,\text{.}$ (30)

This quantity diverges in a similar manner to the predictive partition
function. The expectation value of the kinetic energy, computed from relation
(21) is given by

$\frac{m}{2}\left\langle\dot{\tilde{x}}^{2}(t)\right\rangle_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(T)=x^{\prime\prime}}=\frac{\delta(0)}{2\beta(t)}+\frac{m}{2}\left(\int_{0}^{T}\frac{dt^{\prime}}{\beta(t^{\prime})}\right)^{-2}\frac{\left(x^{\prime\prime}-x^{\prime}\right)^{2}}{\beta^{2}(t)}\,\text{,}$
(31)

which has the divergent term that arises from the uncorrelation between
velocities, but also has a finite part corresponding to the kinetic energy of
the solution $\bar{x}(t)$ of equation (22) with the boundaries given by the
localization condition $\bar{x}(0)=x^{\prime}$ and
$\bar{x}(T)=x^{\prime\prime}$. This solution can be written as

$\bar{x}(t)=\frac{x^{\prime}\int_{t}^{T}dt^{\prime}/\beta(t^{\prime})+x^{\prime\prime}\int_{0}^{t}dt^{\prime}/\beta(t^{\prime})}{\int_{0}^{T}dt^{\prime}/\beta(t^{\prime})}\,\text{,}$
(32)

and corresponds to the most probable path of this model.

The transition probability is found to be finite and corresponds to a Gaussian
distribution with mean equal to $\bar{x}(t)$ and variance consistent with the
localized retrodictive assumptions, as it is zero at boundary times $t=0$ and
$t=T$. The variance or mean square displacement of this distribution is given
by

$\sigma^{2}(t)=\frac{m}{4}\frac{\left(\int_{0}^{t}dt^{\prime}/\beta(t^{\prime})\right)\left(\int_{t}^{T}dt^{\prime}/\beta(t^{\prime})\right)}{\int_{0}^{T}dt^{\prime}/\beta(t^{\prime})}\,\text{.}$
(33)

We see that the integrals in the numerator go to zero for initial and final
times, guaranteeing the certainty in the initial and final values of paths.

A general solution with initial condition $\rho(x|0)=\rho_{0}(x)$ and final
condition $\rho(x|T)=\rho_{T}(x)$, can be written as a convolution

$\rho(x|t)=\int\int\left\langle\delta(\tilde{x}(t)-x)\right\rangle_{\tilde{x}(0)=x^{\prime}}^{\tilde{x}(T)=x^{\prime\prime}}\rho_{0}(x^{\prime})\rho_{T}(x^{\prime\prime})\,dx^{\prime}\,dx^{\prime\prime}\,\text{.}$
(34)

The retrodictive time-dependent dynamics presented in this section do not
correspond to a physical system. The time-dependent distribution obtained
interpolates the evolution of a distribution into another one by assuming the
dynamic is the one of a Brownian motion. Nevertheless, the dynamics shown here
could be useful to account for fluctuations and uncertainty in numerical data
analysis. For example, in systems of few particles, where the solutions often
deviate from the predictive theory.

##  9 Conclusions

The Principle of Maximum Caliber was presented as a comprehensive, general
procedure to deal with non equilibrium statistical mechanics systems. As a
working example, we applied it to study in detail the case of a diffusive
system. The presence of a heat bath is included as a constraint to the
dynamics of the model, in the form of the expectation value of the kinetic
energy of the system. The structure of the prior distribution, inherited from
our knowledge about the system, is responsible for the predictive nature of
the estimations of a given model.

The path-dependent formalism is a clearer representation of macroscopic
systems than the time-dependent statistics framework. Specifically in the
description of the microscopic degrees of freedom involved and their relation
with macroscopic properties. It can be used to write non-equilibrium,
statistical theories in a novel manner that provides insight on the relation
between macroscopic and microscopic dynamics. The Principle of Maximum Caliber
can be interpreted as a link between the dynamic of a macroscopic system and
the specific restrictions that originate that dynamic.

The predictive model presented here is consistent with the results obtained
from the usual time-dependent statistical mechanics; dynamics are irreversible
and the information about the system can be condensed in a transition
probability that can be used to obtain solutions with arbitrary initial
conditions. Kinetic constraints were associated with the thermal dynamics of
Brownian Motion. For arbitrary choice of the parameter $\beta(t)>0$, the mean
square displacement of the macroscopic distribution is always increasing with
time, as it is a sum of positive quantities. In the case where $\beta$ does
not depend on time, we retrieve the normal diffusion process.

Interestingly, the retrodictive model found is also consistent with previously
reported models, as for example the Aharonov-Bergmann-Lebowitz rule [17] for
Quantum Mechanics.

All in all, the Principle of Maximum Caliber, together with the MaxEnt
principle, represent a new, different theoretical effort to build a consistent
theory of equilibrium and non equilibrium statistical mechanics from very few
statements about known information, and deserve to be investigated further.

##  10 Acknowledgments

The authors thank financial support by the National Agency for Research and
Development (Chile) through ANID-PFCHA/Doctorado Nacional/2019-21192159 (I.T.)
and grants FONDECYT 1171127 (G.G.) and FONDECYT 1220651 (S.D.).

<EMAIL_ADDRESS>Tapia<EMAIL_ADDRESS>Gutiérrez
<EMAIL_ADDRESS>Davis dfca p2mcb

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