We have studied dynamics of both internal and external noises-driven dynamical system in terms of information entropy at both nonstationary and stationary states. Here a unified description of entropy flux and entropy production is considered. Based on the Fokker-Planck description of stochastic processes and the entropy balance equation we have calculated time dependence of the information entropy production and entropy flux in presence and absence of nonequilibrium constraint (NEC). In the presence of NEC we have observed extremum behavior in the variation of entropy production as function of damping strength, noise correlation, and non-Gaussian parameter (which determine the deviation of external noise behavior from Gaussian characteristic), respectively. Thus the properties of noise process are important for entropy production.

In recent years the stochastic dynamics [

The outline of the paper is as follows. In Section

We consider a stochastic process in the presence of both internal thermal noise and external noise. The Langevin equation of motion for this process can be written as

Now treating

It is now important to note that one can use the following linear transformation:

By virtue of the above transformation the Fokker-Planck equation (

Now it is important to note that because of nonlinearity in terms of

It describes the variance of the white non-Gaussian noise in the limit

Plot of autocorrelation function (

Thus our present study with the above Fokker-Planck equation will lead to have an exact result when both the internal and the external noises are the Ornstein-Uhlenbeck noise. But the result will be approximate if the external colored noise is non-Gaussian one. It would be close to the exact one as the non-Gaussian noise parameter approaches to unity. Before leaving this part we would like to mention that the random force of internal origin and damping are related through fluctuation-dissipation relation but the external noise is independent of damping. Therefore the stationary state in the present problem is a steady state one. In the next section we will discuss relaxation behavior in terms of entropy production and entropy flux of an external force-driven steady state.

Keeping in mind all the above facts now we introduce the Shanon information measure [

In the next step we define the information entropy flux and entropy production using (

Here

Using the identity

To find the explicit time dependence of the above quantities we then search for Green's function or conditional probability solution [

We will see that by suitable choice of

If we put (

Plot of (

It is now interesting to examine the time dependence of entropy flux and production during the relaxation of small external force-driven steady state. To this end we consider the constant drift

In the next step we use the following time-dependent solution of (

Plot of (

These observations can be explained by simplifying (

We now consider long time behavior of (

Plot of (

Using (

Plot of (

In Figure

Plot of (

In the next step, we have demonstrated the variation of the entropy production as a function of

Plot of (

Finally, in Figure

Plot of (

In conclusion, we have considered the relaxation behavior of a given nonequilibrium state of the thermal broad band noise-driven harmonic oscillator in presence and absence of nonequilibrium constraint. Here we have studied the time dependence of information entropy production and entropy flux based on the Fokker-Planck description of noise process and the entropy balance equation. It includes the following points.

Entropy production monotonically decreases with time to a stationary value in the absence of the non equilibrium constraint (NEC). But in the presence of NEC it first decreases with time and then increases passing through a minimum and finally reaches a limiting value for external Gaussian noise for a given parameter set. But the minimum is going to disappear as the noise behavior deviates more from the Gaussian characteristics.

It is difficult for the non equilibrium constraint to drive the equilibrium state to a steady state as the temperature of the thermal bath increases and the rate of decreases of the entropy production with temperature is fast for external colored Gaussian noise compared to non Gaussian one.

In the presence of NEC we have observed extremum behavior in the variation of entropy production as function of damping strength, noise correlation, non Gaussian parameter (which determine the deviation of external noise behavior from Gaussian characteristic), respectively. Thus the properties of noise process are important for entropy production.

To be mentioned here is that our present calculations are, of course, restricted to the harmonic oscillator (HO). However, insights of this important system usually have a wide impact, as the HO constitutes much more than a mere example. In general Kramers' problem on barrier crossing dynamics is studied analytically by linearization of the nonlinear potential energy function around the fixed points [

Here we have shown that the linear transformation (

Thanks are due to Council of Science and Industrial Research for partial financial support.