Experimental Observation of a Non-Normalizable Boltzmann State

The probability density function of a particle in equilibrium suspended in a fluid follows a Boltzmann distribution, P(x)=exp(-V(x)/k_BT)/Z, where V is the potential energy, k_BT is thermal energy, and Z is the normalizing partition function. However, there are cases where the Boltzmann distribution is not normalizable and the system cannot reach thermodynamic equilibrium. It has recently been shown theoretically and via simulations that, in many instances, a particle coupled to a heat bath approaches in the long time a non-normalizable Boltzmann state, such that P_t(x)=exp(-V(x)/k_BT-x²/4Dt)/Z_t, where D is the diffusion coefficient and Z_t~t^-α [1]. These systems violate the ergodic hypothesis, i.e. the time average of an observable does not converge to the ensemble average in the long time limit. Here, we test these predictions using functionalized polystyrene beads near a charged surface in liquid. The electrostatic potential V(x)~exp(-x/λ_D), with Debye length λ_D, yields a non-normalizable Boltzmann distribution. We track charged particles in 3D near surfaces bearing different surface charge densities. Our experiments reveal a non-normalizable Boltzmann state and ergodicity breaking in agreement with theoretical predictions.
[1] E. Aghion et al., Phys. Rev. Lett. 122, 010601 (2019)