Rigorous foundations for the universal alignment of local thermodynamic arrows of time under weak interactions

The 2012 'Universal Arrow of Time' program of Kupervasser, Nikolić and Zlatić argued that the observed universality of the thermodynamic arrow of time — the fact that all observed regions share one arrow direction — arises because configurations of weakly interacting subsystems with opposite arrow directions are statistically unstable: weak interactions destroy them, with survival probability bounded by $2e^{-N/8}$ in their baker-map toy model. The authors themselves state that for this destruction of opposite time arrows 'a rigorous theorem is still lacking', the analogue for ergodicity being the KAM theorem.

Background

The direction in which macroscopic entropy increases defines the thermodynamic arrow of time. Because the microscopic Hamiltonian dynamics is time-reversal invariant, the existence of the arrow cannot be derived from the dynamics alone; the standard move is to postulate a very special low-entropy initial macrostate of the universe (the Past Hypothesis). Kupervasser, Nikolić and Zlatić (Foundations of Physics 42, 1165 (2012)) addressed a complementary question that does not presuppose a global low-entropy past: why the arrow appears universal, i.e. why interacting regions are not found with arrows pointing in opposite directions. They argued that if two weakly coupled subsystems have opposite arrow directions at some time, the interaction makes the configuration statistically unstable and the subsystems decay toward a common arrow direction; an observer also cannot introspectively observe entropy decrease in himself. Their quantitative support was a toy model of two subsystems, each $N$ particles on the unit square evolved by the baker's map (a simple chaotic, area-preserving map that stretches one coordinate and contracts the other, the standard reversible toy of deterministic chaos), with weak interaction modeled as exchange of positions of nearby particles. In this model the number of solutions consistent with prescribed coarse-grained initial-final conditions is much smaller than for initial conditions alone, and the probability that opposite arrows survive was bounded by $2e^{-N/8}$ using a binomial estimate. The ensemble entropy was modified by coarse-graining — smoothing the phase-space probability density, the continuous analogue of partitioning the square into cells — and the paper compared several such coarse-grainings: the Boltzmann coarsening (based on the single-particle marginal densities), the isotropic coarsening (uniform smoothing of both coordinates), and the Prigogine coarsening (an anisotropic coarse-graining along the contracting direction only, which makes the ensemble-entropy increase law hold without an additional molecular-chaos assumption), the last being singled out as most correct. The interaction also accelerates the decay of correlations, quantified by the Pearson correlation formula $r(m)=2^{-m}/\sqrt{1+4(2^{2m}-1)\sigma^{2}}$, derived under the smallness assumptions $\langle S^{2}\rangle=[(2^{2m}-1)/3]\,\sigma^{2}\ll 1$ and $\sigma^{2}2^{2m}\ll 1$ (which therefore constrain the interaction range $\sigma$ and the number of map steps $m$ jointly).

The broader context is a long-standing debate on whether opposite-arrow regions can coexist. Schulman (Phys. Rev. Lett. 83, 5419 (1999)) exhibited models of two weakly coupled systems with oppositely running thermodynamic arrows, realized through two-time boundary conditions (macrostate constraints imposed at both an initial and a final time); Schulman and da Luz (EPL 106, 10003 (2014)) then asked whether such entities could detect one another and signal. Zeh (Entropy 7, 199 (2005)) argued the opposite: opposite arrows, while logically possible, cannot realistically exist during one and the same epoch — a qualitative version of the instability claim. Goldstein and Tumulka (Class. Quantum Grav. 20, 557 (2003)) used the interplay of opposite microcausal and macrocausal arrows in a relativistic model. None of these works supplies rigorous measure-theoretic statements for generic Hamiltonian systems.

The source paper itself locates the open problem precisely (its Sec. 5): the destruction of opposite arrows is like ergodicity — plausible, valid 'for most practical purposes', with counterexamples (e.g. spin echo, the nuclear-magnetic-resonance phenomenon in which a pulse sequence refocuses dephased spins so that a subsystem's local entropy appears to decrease for an observable time), and difficult to prove. For ergodicity the rigorous touchstone is the Kolmogorov–Arnold–Moser (KAM) theorem — the theory showing that quasi-periodic tori of nearly integrable Hamiltonian systems persist on a large-measure Cantor set, and the paradigm that 'generic' statements in Hamiltonian dynamics can be made rigorous only in a measure-theoretic, non-uniform sense — while for the destruction of opposite arrows, in the authors' words, 'a rigorous theorem is still lacking'.

Problem Statement

Put the interaction-induced alignment mechanism for local thermodynamic arrows of time — proposed by Kupervasser, Nikolić and Zlatić in 'The Universal Arrow of Time' (Foundations of Physics 42, 1165 (2012)) — on rigorous footing beyond their baker-map toy model. The open problem stated by the source itself (its Sec. 5) is the central target below; the further targets are linked directions. Throughout, the systems considered are generic reversible many-body Hamiltonian systems whose subsystems interact weakly (short-range-like interaction, as in the source paper):

1. (Central) Prove or disprove the 'KAM-analogue' theorem: the set (phase-space measure) of microstates realizing prescribed coarse-grained boundary conditions with opposite arrow directions for the interacting subsystems is strongly suppressed in the system size — the source's general claim is qualitative (that this number is 'much smaller' than the number of solutions with only coarse-grained initial or only coarse-grained final conditions), while its toy model establishes the exponential bound $2e^{-N/8}$ (where $N$ is the number of particles in each of the two baker-map subsystems), so a rigorous statement should quantify the suppression (any rigorous quantitative suppression bound in $N$ suffices, with the toy-model exponential bound as the benchmark). The statement must specify the coarse-graining with respect to which the measure is computed (Boltzmann, isotropic, or Prigogine coarsening, as defined above), in what sense the Hamiltonians and boundary conditions are 'generic', and must not smuggle in a low-entropy past hypothesis or a coupling regime that trivializes the conclusion.
2. Quantitative thresholds: determine, at least for explicit classes of systems, the interaction-strength and time-scale thresholds separating decay (alignment) from persistence (coexistence) of opposite-arrow configurations.
3. Mixed-time boundary conditions: for initial data assigned to different subsystems at different times, develop an existence and uniqueness theory (or construct counterexamples) for globally self-consistent solutions of the coupled equations of motion.
4. Finite-$N$ exceptions: characterize quantitatively the systems — spin-echo-type small systems with weak environment coupling and nearly integrable dynamics — in which a local opposite arrow persists for observable times, as exceptions to the asymptotic statement.
5. Validity of the analytic apparatus: delimit systematically the parameter range of the decorrelation formula $r(m)=2^{-m}/\sqrt{1+4(2^{2m}-1)\sigma^{2}}$ under the source's smallness assumptions $\langle S^{2}\rangle=[(2^{2m}-1)/3]\,\sigma^{2}\ll 1$ and $\sigma^{2}2^{2m}\ll 1$ (with $\sigma$ the interaction range and $m$ the number of baker-map steps), and examine to what extent the Prigogine coarsening, which the source argues gives the most correct description of the ensemble-entropy increase law without additional assumptions, remains the appropriate coarse-graining for physical systems beyond the baker map.

The scientific goal of this statement is the central target 1: a rigorous proof — or refutation — of the measure-suppression statement for generic reversible weakly interacting Hamiltonian systems, which is the open problem the source explicitly leaves lacking. Targets 2–5 are linked directions of the same program: results on them constitute partial progress toward a rigorous account of why weakly interacting reversible systems share one thermodynamic arrow direction, but they are not additional completion conditions for the source's open problem, and closing target 1 does not presuppose resolving them.

References

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