The Universal Arrow of Time

Abstract

Statistical physics cannot explain why a thermodynamic arrow of time exists, unless one postulates very special and unnatural initial conditions. Yet, we argue that statistical physics can explain why the thermodynamic arrow of time is universal, i.e., why the arrow points in the same direction everywhere. Namely, if two subsystems have opposite arrow-directions at a particular time, the interaction between them makes the configuration statistically unstable and causes a decay towards a system with a universal direction of the arrow of time. We present general qualitative arguments for that claim and support them by a detailed analysis of a toy model based on the baker’s map.

Appendix: Basic Properties of the Baker’s Map

In this appendix we present some basic properties of the baker’s map. More details can be found, e.g., in [31].

1.1 A.1 Definition of the Baker’s Map

Consider a binary symbolic sequence

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infinite on both sides. Such a sequence defines two real numbers

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The sequence can be moved reversibly with respect to the semicolon in both directions. After the left shift we get new real numbers

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where ⌊x⌋ is the greatest integer less than or equal to x. This map of unit square into itself is called the baker’s map.

The baker’s map has a simple geometrical interpretation presented in Fig. 5. There (a) is the initial configuration and (c) is the final configuration after one baker’s iteration, with an intermediate step presented in (b). The (d) part represents the final configuration after two iterations.

Fig. 5

Geometric interpretation of the baker’s map. (a) Initial configuration. (b) Uniform squeezing in vertical direction and stretching in horizontal direction by a factor of 2. (c) The final configuration after cutting the right half and putting it over the left one. (d) The final configuration after two iterations

1.2 A.2 Unstable Periodic Orbits

The periodic symbolic sequences (0) and (1) correspond to fixed points (x,y)=(0,0) and (x,y)=(1,1), respectively. The periodic sequence (10) corresponds to the period-2 orbit {(1/3,2/3),(2/3,1/3)}. From periodic sequence …001;001… we get {(1/7,4/7),(2/7,2/7),(4/7,1/7)}. Similarly, from …011;011… we get {(3/7,6/7),(6/7,3/7),(5/7,5/7)}.

Any x and y can be approximated arbitrarily well by 0.X ₀…X _n and 0.Y ₀…Y _m , respectively, provided that n and m are sufficiently large. Therefore the periodic sequence (Y _m …Y ₀ X ₀…X _n ) can approach any point of the unit square arbitrarily close. Thus, the set of all periodic orbits makes a dense set on the unit square.

1.3 A.3 Ergodicity, Mixing, and Area Conservation

Due to stretching in the horizontal direction, all close points diverge exponentially under the baker’s iterations. In these iterations, a random symbolic sequence approaches any point of the square arbitrarily close. In general, such an ergodic property can be used to replace the “time” average 〈A〉 by the “ensemble” average

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where dμ(x,y) is the invariant measure and ρ(x,y) is the invariant density for the map. For the baker’s map, ρ(x,y)=1.

Under the baker’s iterations, any region maps into a set of narrow horizontal strips. Eventually, it fills uniformly the whole unit square, which corresponds to mixing. Similarly, reverse iterations map the region into narrow vertical strips, which also corresponds to mixing.

During these iterations, the area of the region does not change. This property is the area conservation law for the baker’s map.

1.4 A.4 Lyapunov Exponent, Shrinking and Stretching Directions

If $x_{0}^{(1)}$ and $x_{0}^{(2)}$ have equal k first binary digits, then, for n<k,

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where Λ=log2 is the first positive Lyapunov exponent for the baker’s map. Consequently, the distance between two close orbits increases exponentially with increasing n, and after k iterations becomes of the order of 1. This property is called sensitivity to initial conditions. Due to this property, all periodic orbits are unstable.

Since the area is conserved, the stretching in the horizontal direction discussed above implies that some shrinking direction must also exist. Indeed, the evolution in the vertical y direction is opposite to that of the horizontal x direction. If $(x_{0}^{(1)},y_{0}^{(1)})$ and $(x_{0}^{(2)},y_{0}^{(2)})$ are two points with $x_{0}^{(1)}=x_{0}^{(2)}$, then

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Hence Λ=−log2 is the second negative Lyapunov exponent for the baker’s map.

1.5 A.5 Decay of Correlations

Since x-direction is the unstable direction, the evolution in that direction exhibits a decay of correlations. The average correlation function C(m) for a sequence x _k is usually defined as

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where $\langle x\rangle=\lim \limits_{n \to\infty}\sum_{k=1}^{n} x_{k}/n$. Correlations can be more easily calculated if one knows the invariant measure μ(x), in which case

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where f ^m(x)=x _m is the function that maps the variable x to its image after m iterations of the map. For the baker’s map dμ(x)=dx, so we can write

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which yields

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For the baker’s map 〈x〉=1/2, so the sum above can be calculated explicitly

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This shows that the correlations decay exponentially with m. The Pearson correlation for the system is given by

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