Mangal C. Mahato

Macroscopic equation of motion in inhomogeneous media: A microscopic treatment

The dynamical evolution of a Brownian particle in an inhomogeneous medium with spatially varying friction and temperature field is important to understand conceptually. It requires to address the basic problem of relative stability of states in nonequilibrium systems which has been a subject of debate for over several decades. The theoretical treatments adopted so far are mostly phenomenological in nature. In this work we give a microscopic treatment of this problem. We derive the Langevin equation of motion and the associated Fokker-Planck equation. The correct reduced description of the Kramers equation in the overdamped limit (Smoluchowski equation) is obtained. Our microscopic treatment may be helpful in understanding the working of thermal ratchets, a problem of much current interest.

Stochastic resonance in periodic potentials.

The phenomenon of stochastic resonance (SR) is known to occur mostly in bistable systems. However, the question of the occurrence of SR in periodic potential systems has not been resolved conclusively. Our present numerical work shows that the periodic potential system indeed exhibits SR in the high-frequency regime, where the linear-response theory yields maximum frequency-dependent mobility as a function of noise strength. The existence of two (and only two) distinct dynamical states of trajectories in this moderately feebly damped periodically driven noisy periodic potential system plays an important role in the occurrence of SR.

Some stochastic phenomena in a driven double-well system

We study the overdamped motion of a Brownian particle in a driven double-well system to understand various physical phenomena observed experimentally. These phenomena include hysteresis, stochastic resonance, and net undirectional motion in a symmetric periodic system. We argue that the area of the hysteresis loop so defined is a good measure of synchronization (with respect to the applied field) of passages between the two wells. We find that stochastic resonance may be relevant even in case of large-amplitude driving field due to a recently discovered phenomena of noise-induced stability of unstable states. We try to find the relation between some of these apparently different phenomena.

Brownian rectifiers in the presence of temporally asymmetric unbiased forces

The efficiency of energy transduction in a temporally asymmetric rocked ratchet is studied. Time asymmetry favors current in one direction and suppresses it in the opposite direction due to which large efficiency approximately 50% is readily obtained. The spatial asymmetry in the potential together with system inhomogeneity may help in further enhancing the efficiency. Fine tuning of system parameters considered leads to multiple current reversals even in the adiabatic regime.

Relation between stochastic resonance and synchronization of passages in a double-well system

We calculate, numerically, the residence times (and their distribution) of a Brownian particle in a two-well system under the action of a periodic, saw-tooth-type, external field. We define hysteresis in the system. The hysteresis loop area is shown to be a good measure of synchronization of passages from one well to the other. We establish a connection between this stochastic synchronization and stochastic resonance in the system.

Stochastic resonance and heat fluctuations in a driven double-well system

We study a periodically driven (symmetric as well as asymmetric) double-well potential system at finite temperature. We show that mean heat loss by the system to the environment (bath) per period of the applied field is a good quantifier of stochastic resonance. It is found that the heat fluctuations over a single period are always larger than the work fluctuations. The observed distributions of work and heat exhibit pronounced asymmetry near resonance. The heat loss over a large number of periods satisfies the conventional steady-state fluctuation theorem.

Deterministic inhomogeneous inertia ratchets

We study the deterministic dynamics of a periodically driven particle in the underdamped case in a spatially symmetric periodic potential. The system is subjected to a space-dependent friction coefficient, which is similarly periodic as the potential but with a phase difference. We observe that frictional inhomogeneity in a symmetric periodic potential mimics most of the qualitative features of deterministic dynamics in a homogeneous system with an asymmetric periodic potential. We point out the need of averaging over the initial phase of the external drive at small frictional inhomogeneity parameter values or analogously low potential asymmetry regimes in obtaining ratchet current. We also show that at low amplitudes of the drive, where ratchet current is not possible in the deterministic case, noise plays a significant role in realizing ratchet current.

Dispersionless motion in a periodically rocked periodic potential.

Recently, dispersionless (coherent) motion of (noninteracting) massive Brownian particles, at intermediate time scales, was reported in a sinusoidal potential with a constant tilt. The coherent motion persists for a finite length of time before the motion becomes diffusive. We show that such coherent motion can be obtained repeatedly by applying an external zero-mean square-wave drive of appropriate period and amplitude instead of a constant tilt. Thus, the cumulative duration of coherent motion of particles is prolonged. Moreover, by taking an appropriate combination of periods of the external field, one can postpone the beginning of the coherent motion and can even have coherent motion at a lower value of position dispersion than in the constant tilt case.

Net particle current in an adiabatically driven unbiased inhomogeneous inertial system in a periodic potential

We extend Riskens matrix continued fraction method (MCFM) to solve the Fokker–Planck equation to calculate the particle current in an inertial symmetric (sinusoidal) periodic potential under the action of a constant force. We consider particle motion in a medium with a small friction coefficient varying periodically in space as the potential but with a finite phase difference . Though the frictional inhomogeneity is considered a less likely candidate to trigger the ratchet effect than the temperature inhomogeneity, for the former leaves the static equilibrium distribution unaffected as opposed to the latter, it results in a finite net algebraic sum of currents with symmetrically time-distributed small applied forces ± |F|. The MCFM results, with very rich and interesting qualitative characteristics, are supported by Langevin dynamic simulations. The effects of different uniform F, temperature T and average friction coefficient γ0 on the performance characteristics of the inhomogeneous ratchet are presented.

Asymmetric motion in a double well under the action of zero-mean Gaussian white noise and periodic forcing

Residence times of a particle in both of the wells of a double-well system, under the action of zero-mean Gaussian white noise and zero-averaged but temporally asymmetric periodic forcings, are recorded in a numerical simulation. The difference between the relative mean residence times in the two wells shows monotonic variation as a function of asymmetry in the periodic forcing and for a given asymmetry the difference becomes largest at an optimum value of the noise strength. Moreover, the passages from one well to the other become less synchronous at small noise strengths as the asymmetry parameter (defined below) differs from zero, but at relatively larger noise strengths the passages become more synchronous with asymmetry in the field sweep. We propose that asymmetric periodic forcing (with zero mean) could provide a simple but sensible physical model for unidirectional motion in a symmetric periodic system aided by a symmetric Gaussian white noise.