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

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.