Active matter and broken equilibrium relations

Consider a one-dimensional telegraph model with direction $\sigma=\pm1$ and speed v0. Direction flips from either sign to the other at rate alpha. This is explicitly a flip rate, not a tumble rate that may choose the same direction again. In the stationary equal-direction ensemble the correlation is $\langle\sigma(t)\sigma(0)\rangle=e^{-2\alpha|t|}$. The factor two follows because either a forward or backward transition relaxes a direction imbalance. A different convention for random reorientation rates would produce a different parameter in the exponential.

Add passive mobility and thermal diffusion through $dx=(v_0\sigma+\mu F)dt+\sqrt{2D_{\mathrm{th}}}\,dW_t$, with independent thermal noise. For the passive bath, $D_{\mathrm{th}}=\mu k_BT$ can still hold. The persistent propulsion is an additional nonequilibrium contribution rather than a change in that bath temperature. At zero force the ensemble mean displacement vanishes when the initial directions are equally probable, but each short trajectory can retain a preferred direction. Zero mean transport therefore does not make the propulsion equivalent to thermal white noise.

Integrating the velocity correlation twice gives the active contribution to displacement variance. At short times it grows as $v_0^2t^2$, expressing persistence, while the independent thermal part grows linearly. At times long compared with the persistence time, the active part becomes linear and adds $v_0^2/(2\alpha)$ to the effective diffusion coefficient. The noncommuting limits are instructive: setting alpha to zero first yields persistent ballistic motion, so the long-time diffusive formula cannot be used at zero flip rate.

Original experiments on chemically propelled colloids observed short-time directed motion and enhanced long-time spreading. That observation supports persistent active descriptions, but does not uniquely select this one-dimensional flip model or its numerical coefficient. Real swimmers rotate in two or three dimensions, interact with walls, and can change speed with fuel concentration. Their orientational correlation and the dimensional diffusion factors must be modeled or measured separately. The formula above is an exact deduction from stated ideal assumptions, not a parameter-free prediction for every experimental active particle.

In this simple model a weak uniform force changes the mean drift by mu times F without altering the propulsion statistics. The ratio $D_{\mathrm{eff}}/\mu$ therefore exceeds the bath thermal energy whenever propulsion is nonzero. Calling that ratio an effective temperature is a limited parametrization of one transport comparison. It does not establish Boltzmann statistics in arbitrary traps, a universal fluctuation–dissipation relation, or standard thermal work identities for all active protocols. Such claims require additional derivations and can fail across observables or timescales.

The coarse-grained propulsion variable also omits chemical fuel and the mechanism that turns fuel into force. Position and direction trajectories alone may not determine the total physical entropy production. A full energetic model must identify reservoirs, fuel reactions, time-reversal rules, and hidden dissipation. Interacting active particles can display collective organization and phase separation in suitable models, but those phenomena do not follow from the single-particle formula alone. Separate exact consequences of a specified stochastic model from empirical observations and broader open questions about real active materials.