Why Does Friction Break Time-Reversal Symmetry?

Friction breaks time-reversal symmetry in the effective equation for a body whose surroundings have been left out of the description. This can be consistent with time-reversal-invariant microscopic dynamics for the larger system.

This note revises my original Chinese answer.

Reversing a damped trajectory

Consider linear drag,

m\ddot x=-b\dot x,\qquad b>0.

If $x(t)$ is a solution, define its reversed path by $x_R(t)=x(-t)$ . Then

\dot x_R(t)=-\dot x(-t),\qquad \ddot x_R(t)=\ddot x(-t).

Substitution gives $m\ddot x_R=+b\dot x_R$ . The reversed path obeys an antidamping equation, not the original drag equation. Equivalently,

\frac{d}{dt}\left(\frac12m\dot x^2\right)=-b\dot x^2\le0.

The model steadily removes the body's kinetic energy. That energy is transferred to degrees of freedom the model does not track.

Reversing the whole system

For microscopic dynamics invariant under time reversal, reversing all relevant momenta produces an allowed reversed trajectory. Any time-reversal-odd external parameters, such as a magnetic field, must also be reversed when required by the theory. It is not enough to reverse the visible body's velocity while leaving its environment in an ordinary thermal state.

During frictional motion, energy and correlations spread into many microscopic degrees of freedom. The exact reversed process requires the corresponding highly coordinated environment to transfer energy back into the body's organized motion. Such states are allowed by reversible dynamics but are not the states normally prepared in a friction experiment.

Loschmidt's objection and statistical assumptions

Loschmidt's reversibility objection exposes why an irreversible kinetic equation cannot follow from reversible mechanics without additional assumptions. The Boltzmann equation uses a molecular-chaos assumption about incoming particles: their pre-collision correlations are neglected. Reversing a correlated post-collision state does not generally preserve that assumption.

The H-theorem is consequently a result for the kinetic equation under its assumptions, not a proof that every microscopic trajectory has increasing entropy. Hamiltonian evolution preserves fine-grained Gibbs entropy when Liouville's theorem applies. Macroscopic entropy increase concerns a coarser description, typicality, and the conditions under which the system was prepared.

Two meanings of reversible

Time-reversal symmetry is a property of dynamics: appropriately reversed trajectories remain allowed. Thermodynamic reversibility is a property of an ideal process: the system and its surroundings can be restored to their initial states without a net change elsewhere. Ordinary friction is thermodynamically irreversible.

Neither definition is simply an equality of forward and reverse transition probabilities. Fluctuation theorems relate probabilities for carefully specified ensembles and protocols. For example, under Crooks's equilibrium-initialization and microscopic-reversibility assumptions,

\frac{P_F(W)}{P_R(-W)}=e^{\beta(W-\Delta F)},

where the two quantities are work probability densities for forward and reversed protocols. This precise relation should not be replaced by a general definition of reversibility. See Crooks's original paper.