Entropy Has No Direction: A Mirror-State Paradox Against Universal Monotonic Entropy Increase and a First-Principles Proof that Constraints Reshape the Entropy Distribution $P_{\infty}(S;\lambda)$

TL;DR

This paper challenges the universal increase of entropy by demonstrating that, under time-reversal invariant dynamics, entropy can be constant or reshaped by constraints, emphasizing the stochastic nature of entropy and its dependence on boundary conditions.

Contribution

It provides a first-principles proof that constraints, not entropy itself, determine the long-term entropy distribution and introduces a mirror-state construction to challenge traditional entropy increase assertions.

Findings

Entropy is a stochastic variable with a distribution shaped by constraints.

The mirror-state construction shows entropy can be constant along trajectories.

Constraints and boundary conditions, not entropy, are manipulated to achieve system behaviors.

Abstract

We revisit textbook claims that entropy must increase and show that, under time-reversal invariant microscopic dynamics, no universal trajectory-wise or statistical assertion that the coarse-grained entropy $S(t)$ is non-decreasing can hold. The core is a mirror-state construction: for any microstate $A$ one constructs its time-reversed partner $B$ (momenta inverted); requiring $S(t)$ to be non-decreasing for both $A$ and $B$ forces every time to be a local minimum of $S$ and hence makes $S(t)$ constant along the trajectory. The consistent picture is that entropy is a stochastic variable described by a probability distribution $P(S)$ whose shape depends on constraints and boundary conditions; entropy-based regularities are emergent summaries of constraint-dependent microscopic dynamics, and in practice it is constraints and boundaries -- not entropy itself -- that one manipulates to achieve mixing, separation, or self-organization. Working with Boltzmann (coarse-grained) entropy on the energy shell, we then derive from first principles how constraints reshape the long-time entropy distribution $P_{\infty}(S;\lambda)$ by altering the invariant measure through changes in the Hamiltonian and/or the accessible phase space. In the microcanonical setting we obtain a sharp criterion: the \emph{only} way $P_{\infty}^{(E)}(S;\lambda)$ can remain the same up to translation is when all accessible macrostate volumes are scaled by a common factor; otherwise the distribution changes structurally. We connect this framework to experiments on asymmetric nanopores and molecular gates, to macroscopic examples from civil engineering (windbreak forests, dikes, vortex suppression, traffic-flow control), and to natural phenomena such as lightning guided to lightning rods, snowflake and mineral-veil growth, and the sudden crystallisation of supercooled water.