Brownian Motion In Periodic Potentials Research Articles

Probability distribution of Brownian motion in periodic potentials

We calculate the probability distribution function (PDF) of an overdamped Brownian particle moving in a periodic potential energy landscape $U(x)$. The PDF is found by solving the corresponding Smoluchowski diffusion equation. We derive the solution for any periodic even function $U(x)$, and demonstrate that it is asymptotically (at large time $t$) correct up to terms decaying faster than $\sim t^{-3/2}$. As part of the derivation, we also recover the Lifson-Jackson formula for the effective diffusion coefficient of the dynamics. The derived solution exhibits agreement with Langevin dynamics simulations when (i) the periodic length is much larger than the ballistic length of the dynamics, and (ii) when the potential barrier $\Delta U=\max(U(x))-\min(U(x))$ is not much larger than the thermal energy $k_BT$.

Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials.

Using Brownian motion in periodic potentials V(x) tilted by a force f, we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation q, and in order to limit the output fluctuation within a relative uncertainty ε, at least 2k_{B}T/ε^{2} of heat must be dissipated. Our model shows that this bound is attained not only at near-equilibrium [f≪V^{'}(x)] but also at far-from-equilibrium [f≫V^{'}(x)], more generally when the dissipated heat is normally distributed. Furthermore, the energetic cost is maximized near the critical force when the barrier separating the potential wells is about to vanish and the fluctuation of Brownian particles is maximized. These findings indicate that the deviation of heat distribution from Gaussianity gives rise to the inequality of the uncertainty relation, further clarifying the meaning of the uncertainty relation. Our derivation of the uncertainty relation also recognizes a bound of nonequilibrium fluctuations that the variance of dissipated heat (σ_{q}^{2}) increases with its mean (μ_{q}), and it cannot be smaller than 2k_{B}Tμ_{q}.

Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency

We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale.

The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

The continued-fraction method to solve classical Fokker--Planck equations has been adapted to tackle quantum master equations of the Caldeira--Leggett type. This can be done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).

Bistability effects of the brownian motion in periodic potentials

In the stationary state the equation of motion for particles moving in a periodic potential has two solutions, a locked one and a running one, for low and intermediate damping constants and for suitable external forces. The effect of an additional Langevin force to this bistable behaviour is investigated. For finite noise strength, the mobility depends continuously on the external force, whereas in the limit of vanishing strength of the noise force one gets a sharp transition between the locked and the running solution at a critical external force. This critical force is calculated exactly in the low friction limit and approximately for intermediate friction constants. Furthermore the temperature dependence for various forces including the critical one is shown in the low friction limit.

Brownian motion in periodic potentials; nonlinear response to an external force

The Brownian motion of particles in a periodic potential in response to a constant external force is investigated. By expanding the distribution function into Hermite-functions and into a Fourier-series, the Fokker-Planck-equation is transformed into a set of coupled equations for the expansion coefficients. These equations are solved by a continued fraction method for matrices. This continued fraction for the matrices converges for large, intermediate and even for very small damping constants. The mobility, the kinetic and potential energy for various damping constants and external forces are given for a cos-potential. The current-voltagecharacteristic of the Josephson tunneling junction is also shown.

Brownian motion in periodic potentials in the low-friction-limit; nonlinear response to an external force

The Brownian motion of particles in a periodic potential in response to a constant external force is investigated in the low-friction-limit. By introducing an energy and a space variable and by making a proper self consistent ansatz, the Fokker-Planck equation is solved in the stationary state for small friction constants. We found out that, for fixed force to friction ratios, the mobility times the friction constant is a linear function of the square root of the friction constant if the damping is small enough. Explicit results for this linear function are presented for a cosine potential and compared to previous results.