Abstract
We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker–Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker–Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.
Highlights
Systems with interactions that vanish at long distances are ubiquitous in nature, ranging from charged particles to cosmological objects
We will focus our attention on one dimensional systems, where x represents the coordinate of the particle; the stagnation of the dynamics in a well that is asymptotically flat, for a particle that is described by the Langevin dyanmics, is a general phenomenon that is found in higher dimensions
What is remarkable is that certain aspects of standard statistical physics and thermodynamics can still be applied through a suitable regularization of Peq ( x ) and observable averages, as we have shown in previous work [1], where we developed a general formalism for the problem
Summary
Systems with interactions that vanish at long distances are ubiquitous in nature, ranging from charged particles to cosmological objects. E, and the entropy S , as well as dynamical ones, e.g., the mean square displacement (MSD), attain a time-independent value and the virial theorem approximately holds, which immediately raises the question about the possibility of using Boltzmann-Gibbs statistics This is nontrivial, because the expression for the equilibrium probability, for instance, in one dimension, defined as Peq ( x ) = Z1 e−V ( x)/(k B T ) , where k B is the Boltzmann constant,. What is remarkable is that certain aspects of standard statistical physics and thermodynamics can still be applied through a suitable regularization of Peq ( x ) and observable averages, as we have shown in previous work [1], where we developed a general formalism for the problem This was based on scaling solutions of the Fokker–Planck equation (FPE) for the probability density function (PDF) P( x, t), and alternatively on finite-box solutions.
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