Kramers problem for a dimer: Effect of noise correlations

The Kramers problem for a dimer in a bistable piecewise linear potential is studied in the presence of correlated noise processes. The effect of such a correlation is to redistribute the thermal power between the dynamical degrees of freedom, and this leads to significant deviations in the dynamics of the system from the case of independent noise processes. The distribution of first passage times from one minima to the basin of attraction of the other minima is found to have exponentially decaying tails with the parameter dependent on the amount of correlation and the coupling between the particles. The strong coupling limit of the problem is analyzed using adiabatic elimination, where it is found that the initial probability density relaxes towards a stationary value on the same time scale as the mean escape time when the noise intensity of the system is low. For higher noise fluctuations, the relaxation towards the stationary state is slower in comparison to escape times. In the extreme limit of perfect anticorrelation, the random dynamical system behaves as a deterministic system in a steady state in which the center of mass starting from the unstable maxima moves down the hill and gets trapped in the potential minima. The implications for polymer dynamics in a potential are discussed.

Figure 1

Sample trajectories of the dimer $x_1$ (black solid) and $x_2$ (red dash) in the piecewise linear bistable potential $U$, with the particles interacting simple harmonically for the initial conditions $(x_c,x_r)=(-1.0,0.02)$ for different values of spring constant $k$ and noise correlation $\rho$ for noise intensity $D=0.25$. The time $t$ in the figure is in multiples of ${10}^3$.

Figure 2

Cumulative distribution of the first passage times $\tau$ of the center of mass starting in the left well to the absorbing boundary at $x_c=0$. The distribution has exponentially decaying tails with parameter $\langle \tau \rangle$. The effect of the coupling between the two monomers is evident on the nature of the distribution: for nearly independent movement of the particles ($k=0.01$) (a) the distributions trace each other for different values of $\rho$. When the coupling between the monomers increases, the mean first passage time $\langle \tau \rangle$ is found to decrease with increasing $\rho$ as observed for $k=0.1$ (b) and $k=1$ (c). The distributions are calculated using 500 000 data points for noise intensity $D=0.25$.

Figure 3

Rate of escape $R$(solid lines) and the rate of relaxation $1/T$(dashed lines) against the correlation $\rho$. The numerical results for the rate of escape are calculated from the mean first passage time of the dimer from the potential minima at $x_c=-1$ to the basin of attraction of the other minima. The numerical values are obtained by the solution of Eq. (9) by averaging over 10 000 ensembles. The formula for the rate of escape $R$ is used from Eq. (8), with $1/R=D_{\rho }(e^{2/D_{\rho }}-1)-1$, as the approximate form by Kramers holds only in the limit of small noise intensity and is also reflected in the deviation between the solid and dashed lines.