On the Statistical Viewpoint Concerning the Second Law of Thermodynamics—A Reminder on the Ehrenfests’ Urn Model

Abstract

In statistical thermodynamics the 2nd law is properly spelled out in terms of conditioned probabilities. As such it makes the statement that “entropy increases with time” without preferring a time direction. In this paper we wish to explain and illustrate this statement in terms of the Ehrenfests’ urn model in a way that hopefully adds some clarifying aspects concerning the role of time-conditioned probabilities. We will relate past- and future-conditioned probabilities through Bayes’ rule, which allows us to explicitly state what is meant by time-reversal invariance in this context.

7 Appendix

In this Appendix we collect some elementary notions of probability theory, adapted to our specific example.

The space of elementary events¹¹ is $\Omega =\{0,1,\dots ,N\}$. By

$$\begin{aligned} \mathcal {X}:= & {} \bigl \{X:\Omega \rightarrow \mathbb {R}\bigr \}\,, \end{aligned}$$

(57)

$$\begin{aligned} \mathcal {W}:= & {} \bigl \{W:\Omega \rightarrow \mathbb {R}_{\ge 0}\mid \sum _{z\in \Omega }W(z)=1\bigr \}\,, \end{aligned}$$

(58)

we denote the sets of random variables and probability distributions respectively, where $\mathcal {W}\subset \mathcal {X}$. The map $\mathcal {X}\rightarrow \mathbb {R}^{N+1}$, $X\mapsto (X(0),X(1),\cdots ,X(N))$ defines a bijection which allows us to identify $\mathcal {X}$ with $\mathbb {R}^{N+1}$. This identifies $\mathcal {W}$ with the N-simplex

$$\begin{aligned} \Delta ^N:=\bigl \{(W(0),\cdots ,W(N))\in \mathbb {R}^{N+1}\mid W(z)\ge 0,\, \sum _{z}W(z)=1\bigr \} \subset \mathbb {R}^{N+1}\,. \end{aligned}$$

(59)

Its boundary, $\partial \Delta ^N$, is the union of all $(N-K)$-simplices:

$$\begin{aligned} \Delta ^{i_1\cdots i_K}:=\bigl \{(W(0),\dots ,W(N))\in \Delta ^N\mid 0=W(i_1)=\cdots =W(i_K)\bigr \} \,, \end{aligned}$$

(60)

for all K. Its interior is ${\displaystyle {\mathop {\mathcal {W}}^{\circ }}}:=\mathcal {W}-\partial \mathcal {W}$, so that $W\in {\displaystyle {\mathop {\mathcal {W}}^{\circ }}}\Leftrightarrow W(z)\not =0\forall z$.

Expectation value ${E}$, variance ${V}$, and standard deviation ${S}$ are functions $\mathcal {X}\times \mathcal {W}\rightarrow \mathbb {R}$, defined as follows:

$$\begin{aligned}&{E}:\mathcal {X}\times \mathcal {W}\rightarrow \mathbb {R},\quad {E}(X,W):=\sum _{z\in \Omega }X(z)W(z) \,, \end{aligned}$$

(61)

$$\begin{aligned}&{V}:\mathcal {X}\times \mathcal {W}\rightarrow \mathbb {R}_{\ge 0},\quad {V}(X,W):={E}((X-\langle X\rangle )^2,W)={E}(X^2,W)-{E}^2(X,W)\,, \end{aligned}$$

(62)

$$\begin{aligned}&{S}:\mathcal {X}\times \mathcal {W}\rightarrow \mathbb {R}_{\ge 0},\quad {S}(X,W):=\sqrt{{V}(X,W)}\,, \end{aligned}$$

(63)

where in (62) $\langle X\rangle$ simply denotes the constant function $\langle X\rangle :z\mapsto {E}(X,W)$, and ${E}^2(X,W):=[{E}(X,W)]^2$. In the main text we also write ${E}(X,s)$ if the symbol s uniquely labels a point in $\mathcal {W}$, like $s=\text {ap}$ for the a priori distribution (1), or $E(X,t_i)$ for the distribution $W_i$ at time $t_i$.