Ergodic Theory, Interpretations of Probability and the Foundations of Statistical Mechanics
Introduction
This paper investigates the role that ergodic theory can play in the foundations of equilibrium statistical mechanics. Historically, the mathematical theory of ergodic theory developed in the early twentieth century in close connection with foundational issues in statistical mechanics. Its original role was to establish a connection between ensemble functions (phase averages) and properties of individual systems (time averages). Whether it gives a valid justification for the use of ensembles in statistical physics is however much disputed.
In this paper I will discuss three ergodic approaches that differ from the standard ergodic approach. Two of them are meant as support of specific interpretations of probability in statistical mechanics, namely of the time average interpretation, and the personalist interpretation, respectively. Both are advocated by Von Plato and Guttmann (Von Plato 1988, Von Plato 1989; Guttmann, 1999). The third approach, originally put forward by Malament and Zabell and elaborated by Vranas, is aimed at the same goal as the standard ergodic approach, namely to explain the success of the phase averaging method, but uses a different line of argument (Malament and Zabell, 1980; Vranas, 1998). All three approaches concern the foundations of equilibrium theory, and thus have nothing to do with the issue of irreversibility, which is yet another area where ergodic theory may have a role.
A second aim of this paper is to investigate the relevance of the interpretation of probability to the above-mentioned ergodic approaches. It is natural to associate ergodic theory with objective interpretations of probability. This is because with the use of ergodic theory a connection can be established between probability measures on the one hand, and objective features of real world systems on the other. However, ergodic theory can be useful also with other interpretations of probability, as is shown by the use of the ergodic decomposition theorem as support for the personalist interpretation of probability.
The interpretation of probability and the ergodic approaches are clearly connected in the sense that ergodic theory has been invoked as putative support for two distinct interpretations of probability: the time average interpretation, and the personalist interpretation. Another sense in which the interpretation of probability is relevant is, I will argue, that the plausibility of certain assumptions that are made in the mentioned ergodic approaches depends on the interpretation of probability.
In order to illustrate both the diversity of foundational roles of ergodic theory and the importance of the interpretation of probability, I will single out one particular assumption about the probability distribution that plays a role in all three ergodic approaches, namely stationarity. A probability distribution is stationary if it is constant at all fixed points in phase space. This is usually taken to reflect the fact that the system described by the probability distribution is in equilibrium. However, as I will argue, stationarity of the ensemble is not the right way to account for the system being in equilibrium. I will present another, weaker condition on the probability distribution as representing thermal equilibrium, and will investigate what effect this modification has on the three ergodic approaches.
This paper is structured as follows. In Section 2, I will review the traditional role of ergodic theory in the foundations of statistical mechanics and the reasons why it is problematic. In Section 3, I will discuss the three alternative uses of ergodic theory in the foundations of equilibrium statistical mechanics. In Section 4, I will highlight the notion of stationarity, and argue for a weakened account of equilibrium.
Section snippets
Standard role of ergodic theory in the foundations of statistical mechanics
The traditional use of ergodic theory in the foundations of equilibrium statistical mechanics is to make a connection between the ensembles used in statistical mechanics and properties of single systems. More specifically, ergodic theory is invoked to solve the ergodic problem, which is to demonstrate the equality of infinite time averages and phase averages, i.e. expectation values with respect to the microcanonical measure on phase space. A related goal is to explain the success of
A Plurality of Ergodic Approaches
In this section I will discuss three different roles that ergodic theory can play in the foundations of equilibrium statistical mechanics. The first two are as putative support for particular interpretations of probabilities, namely the time average interpretation and the personalist interpretation, respectively. The third is aimed at exactly the same goal as the traditional ergodic approach, namely to provide an explanation of the success of the microcanonical phase averaging method; it
Does stationarity represent equilibrium?
As an illustration of the differences between the ergodic approaches outlined above, and of the importance of the interpretations of probability, let us look at the notion of stationarity in more detail. It is clear that stationarity plays an important role in all three ergodic approaches. In Malament and Zabell's scheme stationarity is one of the main assumptions. The ergodic decomposition theorem is a representation theorem for stationary measures. Finally, time averages are stationary by
References (17)
- et al.
Why Ergodic Theory Does Not Explain the Success of Equilibrium Statistical Mechanics
British Journal for Philosophy of Science
(1996) - Ehrenfest, P. and Ehrenfest-Afanassjewa, T. (1912) The Conceptual Foundations of the Statistical Approach in Mechanics...
The Concept of Probability in Statistical Physics
(1999)- Jaynes, E. T. (1967) ‘Foundations of Probability Theory and Statistical Mechanics’, in M. Bunge (ed.), The Delaware...
Mathematical Foundations of Statistical Mechanics
(1949)- Lanford, O. E. (1973) ‘Entropy and Equilibrium States in Classical Statistical Mechanics’, in A. Lenard (ed.),...
Discussion: Malament and Zabell on Gibbs Phase Averaging
Philosophy of Science
(1989)- et al.
Why Gibbs Phase Averages work—The Role of Ergodic Theory
Philosophy of Science
(1980)