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arXiv:cond-mat/0410063

The Backwards Arrow of Time of the Coherently Bayesian Statistical Mechanic

Cosma Rohilla Shalizi

Published Oct 4, 2004• 4 citations
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Abstract

Many physicists think that the maximum entropy formalism is a straightforward application of Bayesian statistical ideas to statistical mechanics. Some even say that statistical mechanics is just the general Bayesian logic of inductive inference applied to large mechanical systems. This approach identifies thermodynamic entropy with the information-theoretic uncertainty of an (ideal) observer's subjective distribution over a system's microstates. In this brief note, I show that this postulate, plus the standard Bayesian procedure for updating probabilities, implies that the entropy of a classical system is monotonically non-increasing on the average -- the Bayesian statistical mechanic's arrow of time points backwards. Avoiding this unphysical conclusion requires rejecting the ordinary equations of motion, or practicing an incoherent form of statistical inference, or rejecting the identification of uncertainty and thermodynamic entropy.

DOI
10.48550/arxiv.cond-mat/0410063

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Many physicists think that the maximum entropy formalism is a straightforward application of Bayesian statistical ideas to statistical mechanics. Some even say that statistical mechanics is just the general Bayesian logic of inductive inference applied to large mechanical systems. This approach identifies thermodynamic entropy with the information-theoretic uncertainty of an (ideal) observer's subjective distribution over a system's microstates. In this brief note, I show that this postulate, plus the standard Bayesian procedure for updating probabilities, implies that the entropy of a classical system is monotonically non-increasing on the average -- the Bayesian statistical mechanic's arrow of time points backwards. Avoiding this unphysical conclusion requires rejecting the ordinary equations of motion, or practicing an incoherent form of statistical inference, or rejecting the identification of uncertainty and thermodynamic entropy.
We examine the fundamental aspects of statistical mechanics, dividing the problem into a discussion purely about probability, which we analyse from a Bayesian standpoint. We argue that the existence of a unique maximising probability distribution {p(j∣K)}\{p(j\vert K)\}{p(j∣K)} for states labelled by jjj given data KKK implies that the corresponding maximal value of the information entropy σ({(pj∣K)})=−∑j(pj∣K)ln⁡(pj∣K)σ(\{(p_j\vert K)\}) = -\sum_j (p_j \vert K)\ln{(p_j\vert K)}σ({(pj​∣K)})=−∑j​(pj​∣K)ln(pj​∣K) depends explicitly on the data at equilibrium and on the Hamiltonian of the system. As such, it is a direct measure of our uncertainty about the exact state of the body and can be identified with the traditional thermodynamic entropy. We consider the well known microcanonical, canonical and grand canonical methods and ensure that the fluctuations about mean values are generally minuscule for macroscopic systems before identifying these mean values with experimental observables and thereby connecting to many standard results from thermodynamics. Unexpectedly, we find that it is not generally possible for a quantum process to be both isentropic and reversibly adiabatic. This is in sharp contrast to traditional thermodynamics where it is assumed that isentropic, reversible adiabatic processes can be summoned up on demand and easily realised. By contrast, we find that linear relations between pressures PjP_jPj​ and energies EjE_jEj​ are necessary and sufficient conditions for a quasi-static and adiabatic change to be isentropic, but, of course, this relationship only holds for a few especially simple systems, such as the perfect gas, and is not generally true for more complicated systems. By considering the associated entropy increases up to second order in small volume changes we argue that the consequences are in practice negligible.
What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of entropy as it appears in the statistical interpretation of the second law of thermodynamics. At the same time, the subjective information-theoretical interpretation of probability, and hence entropy, is a standard viewpoint in the foundations of statistical mechanics. In this article, it is argued that the subjective interpretation is incompatible with the philosophical point of view that the arrow of time is a fundamental property of Nature. The subjectivist can only uphold this philosophy if the role played by the second law of thermodynamics in defining time's arrow is abandoned.
What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of entropy as it appears in the statistical interpretation of the second law of thermodynamics. At the same time, the subjective information-theoretical interpretation of probability, and hence entropy, is a standard viewpoint in the foundations of statistical mechanics. In this article, it is argued that the subjective interpretation is incompatible with the philosophical point of view that the arrow of time is a fundamental property of Nature. The subjectivist can only uphold this philosophy if the role played by the second law of thermodynamics in defining time's arrow is abandoned.
What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of entropy as it appears in the statistical interpretation of the second law of thermodynamics. At the same time, the subjective information-theoretical interpretation of probability, and hence entropy, is a standard viewpoint in the foundations of statistical mechanics. In this article, it is argued that the subjective interpretation is incompatible with the philosophical point of view that the arrow of time is a fundamental property of Nature. The subjectivist can only uphold this philosophy if the role played by the second law of thermodynamics in defining time's arrow is abandoned.
What is the physical origin for the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of disorder, or entropy, as it appears in the statistical interpretation of the second law of thermodynamics. In this article, it is argued that the philosophical point of view that the arrow of time is a fundamental property of Nature is incompatible with the subjective information-theoretical interpretation of probability and entropy in statistical mechanics, where they arise due to the ignorance of the observer. This conclusion, however, relies on the premise that time's arrow has a statistical mechanical origin. If this assumption is abandoned, then there is no conflict between the belief in the fundamental character of time's arrow and the belief in the subjective information-theoretical interpretation.
What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of disorder, or entropy, as it appears in the statistical interpretation of the second law of thermodynamics. In this article, it is argued that the philosophical point of view that the arrow of time is a fundamental property of Nature is incompatible with the subjective information-theoretical interpretation of probability and entropy in statistical mechanics, where they arise due to the ignorance of the observer. This conclusion, however, relies on the premise that time's arrow has a statistical mechanical origin. If this assumption is abandoned, then there is no conflict between the belief in the fundamental character of time's arrow and the belief in the subjective information-theoretical interpretation.