Analytical Solution to Transport in Brownian Ratchets via the Gambler’s Ruin Model

Abstract

We present an analogy between the classic Gambler's Ruin problem and the thermally-activated dynamics in periodic Brownian ratchets. By considering each periodic unit of the ratchet as a site chain, we calculated the transition probabilities and mean first passage time for transitions between energy minima of adjacent units. We consider the specific case of Brownian ratchets driven by Markov dichotomous noise. The explicit solution for the current is derived for any arbitrary temperature, and is verified numerically by Langevin simulations. The conditions for vanishing current and current reversal in the ratchet are obtained and discussed.

Efficiency of rejection-free methods for dynamic Monte Carlo studies of off-lattice interacting particles

“…The standard dynamic MC algorithm for particle systems involves the following six iterative steps, with one iteration being called a MC step (MCS). We have used the term dynamic MC for this algorithm, rather than the term time-quantified MC used in [13,14], since it has been used previously [10,11,12,15]. It is important to remember that in dynamic MC the time in MCS is proportional to the physical time in seconds [9,10,11,12,13,14].…”

“…We have used the term dynamic MC for this algorithm, rather than the term time-quantified MC used in [13,14], since it has been used previously [10,11,12,15]. It is important to remember that in dynamic MC the time in MCS is proportional to the physical time in seconds [9,10,11,12,13,14]. The algorithm satisfies detailed balance.…”

“…However, there are other instances where the physical time evolution of the system is desired while rate constants might be unavailable (for example perhaps transition state theory does not apply or an important complicated multi-particle motion might be difficult to conceptualize or calculate). In such cases the physical time development may sometimes still be derived from the underlying physical system, for example by studying the underlying quantum mechanism for time development [9,10,11,12], or devising a method equiva-lent to the time development of the underlying equations [13,14].…”

Deciding when to quit the gambler's ruin game with unknown probabilities

“…Despite such a long history in the literature, the gambler's ruin is still an actively studied problem [11,12,13,14,15,16,17,18,19,20]. It is also used as a model for diverse applied problems, such as business risk [21,22,23], quantum mechanics [24,25], anomaly detection [26], material properties [27,28,29], or yet other physical [30,31], biological [32,33], and even social phenomena [34]. The gambler's ruin has recently been associated with a survival version of multiarmed bandits [35,36,37], which in turn is an essential model for studying sequential decision and learning problems [38].…”

Quantum ratchet in two-dimensional semiconductors with Rashba spin-orbit interaction

“…For example, the thermal diffusion of particles can be ‘chopped’ by a time-modulated asymmetrical potential barrier and this leads to a directed motion of particles 8 9 . Alternatively, net flow of particles across asymmetrical potential barrier can also be driven by dichotomous Markov noise 10 11 . Such devices belongs to the class of classical Brownian ratchets since the ratchet current originates from the classical Brownian diffusion of particles.…”