Steady-state currents in stochastic ratchets: Analytical results (Q2722389)

In this conference article the authors find explicit formulas for the steady-state current in stochastic dynamical systems with asymmetric periodic potential driven by time-dependent external forces -- the so-called stochastic ratchets. They consider the so-called liquid (high damping) phase, where the diffusive dynamics of a particle in a thermal bath is given by the first-order Smoluchowski equation, in two different models for the one-dimensional motion with position \( x(t) \): NEWLINENEWLINENEWLINE(i) ``fluctuating force'' (additive ratchet) \( \dot{x}(t)=-U'(x)+\sqrt{2D}\dot{w}(t)+\xi(t),\)NEWLINENEWLINENEWLINE(ii) ``fluctuating barrier'' (multiplicative ratchet) \( \dot{x}(t)=-U'(x)\xi(t)+\sqrt{2D}\dot{w}(t)\),NEWLINENEWLINENEWLINEwhere \( U \) is the deterministic potential, \( \dot{w} \) the standard Gaussian white noise, and \( \xi \) is the stochastic process for the external force. These problems inherit three time scales; the first is the force fluctuation \( \xi(t) \), denoted by \( \tau\); the second, \( 1 /D \), the diffusive scale and the third is associated with the deterministic dynamics, i.e., this time is proportional to \( \max[1/U''(x)]^{1/2} \). In determining the steady-state current the interplay between the three time scales is the key point. The authors consider ``rapid jumps'', i.e., they assume that \( \tau \) is small compared with the diffusive scale \( 1/D \) and two cases of potential: \( |U''|\) bounded (smooth potential) and \( |U''|\) unbounded (sharp potential). Because of the amount of previous work on numerical studies of ratchets with piecewise linear (i.e., sharp) potentials they provide some motivation for their analytical study of the problem. The authors consider as sharp potential the simplest such potential over the entire real line, which is piecewise linear and periodic. As external force \( \xi(t) \) they take the zero mean non-symmetric telegraph process with values \( b \) and \( -a \) and exponentially distributed switching times with parameter \( \lambda_1 \) and \( \lambda_2 \), such that \(\lambda_1+\lambda_2=1/\tau \). Properties of \( \xi \) are given by \( a,\tau \) and \( h=b-a\). NEWLINENEWLINENEWLINEThe steady-state transition probability satisfies the forward master equation, where one can derive an equation for the steady-state current. The authors analyze the dependence of the steady-state current on the time-scales via analytical methods and by performing symbolic calculations using Maple V 400 MHz Pentium II Linux-based system, and they get the following results: With no jumps there is no steady-state current. In model (i) the steady-state current is transcendentally small if \( D=0\), \(\tau\ll 1\), and it increases transcendentally large for nonzero \( D \). Moreover they derive the value of \( D \) which maximizes the current for given \( \tau \). In model (ii) there is no steady-state in the case \( D=0,\tau\ll 1\). Nonzero \(D\) ensures steady-state with current independent of \( D \) and proportional to \( \tau\). The authors show that the current in the case of sharp potential is asymptotically larger than in the case of the smooth potential.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00031].