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In this paper, we define the harmonic oscillator with random damping in non-Markovian thermal bath. This model represents new version of the random oscillators. In this side, we derive the overdamped harmonic oscillator with multiplicative colored noise and translate it into the additive colored noise by changing the variables. The overdamped harmonic oscillator is stochastic differential equation driving by colored noise. We derive the change in the total entropy production (CTEP) of the model and calculate the mean and variance. We show the fluctuation theorem (FT) which is invalid at any order in the time correlation. The problem of the deriving of the CTEP is studied in two different examples of the harmonic potential. Finally, we give the conclusion and plan for future works.

In the 1980s, studies of linear and non linear oscillators were extended to the case of colored noise driving force. Many applications of the random damping in Markovian thermal bath include water waves influenced by turbulent wind field, the Ginzburg-Landau equation with a convective term, mean flow passing through a region under study, open flows of liquids, dendritic growth, chemical waves and motion of vortices [

In this section, we define the harmonic oscillator with random damping in non-Markovian thermal bath. We de- rive the stochastic differential equation (SDE) driving by the multiplicative colored noise and translate it into the additive colored noise by changing variables in overdamped approximation and its stochastic treatment. Our model can be defined as,

where

where

and read the Equation (1) as,

In overdamped approximation , the Equation (1) become,

taking the time derivative of Equation (3) we get,

Substituting Equation (5) in (6), one can obtain,

let

By using power series at first order in the noise

let

cative colored noise. To translate Equation (9) from multiplicative colored noise into additive ,we must divided Equation (9) by

let,

then the translation [

then the Equation (10) is,

Equation (13) represent SDE driving by additive colored noise. The Fokker Planck equation [

where the initial condition

we take the time derivative into above equation, we have,

Assume that

where

To obtain the initial distribution function we must assume that

mean (

where the initial distribution is exponential distribution. Then the marginal probability of the particle’s position is,

also the distribution of y is exponential distribution , and note that y and

In this section, we change variables in the time dependent driving force from x into y which is defined as

where

Substituting Equation (22) in Equation (21), one can obtain,

Equation (23) represent first new formula of harmonic potential in y. The change in new harmonic potential can be defined as,

where we assume that

where the work can be computed [

The mean of the work is calculated as,

where the quantity

Putting Equation (28) inside Equation (27), we get,

the variance of the work can be calculated as,

where the quantity

Substituting Equation (31) inside Equation (30). one can obtain,

The change in the environment entropy

The change in entropy of the system

Now, we can find the CTEP

where

we must calculate the following quantities

where

and,

note here

before we find the variance of the CTEP, we make some the following assumptions, let

we must calculate the following quantities

note here

and,

Putting Equation (32) and an above quantities’s values in Equation (41), one can obtain,

At zero order in time correlation

where

and variance,

where

In this section , we change variables in the time dependent driving force from x into y which is defined as

where

Equation (52) represent second new formula of harmonic potential in y .

where we assume that

The mean of the work is calculated as,

and the variance of the work is,

We note that, the mean of the work in first and second example are different, while, the variance is equal. The change in the environment entropy

where

The variance of the

since the variance of the

then we note, the variance of the change in entropy of the environment is same in tow examples, while the mean is different. Now, we can calculate

The mean of the

The variance of the

At zero order in time correlation , the CTEP

the mean is,

and variance is,

where

ples are different. From Equations (59) and (60), we conclude that the variance of the CTEP is the same in two examples and at any order in time correlation and also we can not find any linear relation between the mean and variance of the CTEP at any order in time correlation , while ref. [

In this section, we show the fluctuation theorem (FT) is invalid at any order in time correlation whether the distribution function of the change in the total entropy production (CTEP) is Gaussian or non Gaussian. We study the distribution function of the CTEP with respect first example , because any example chosen no problem. We base on the relation between the moments and cumulants to find the distribution function of the CTEP which is defined as,

where

from above equation, we find that the

The other perspective, at any order in u, this perspective is studied in [

Based on Equation (70), we note that ,at any order in time correlation, the FT in our work is invalid, while at zero order in time correlation, the FT in [

In this letter, we defined the harmonic oscillator with random in non-Markovian thermal bath and we derived the SDE driving by multiplicative colored noise. By changing variables, we translated SDE from multiplicative colored noise into additive colored noise to become the calculations easier. Under the new formulas of the har- monic potential in the two examples, we derived the change in the total entropy production (CTEP) of the our model and calculated the mean and the variance. By comparing our results in the two examples, we found the variances of the CTEP are the same while the means are different. At zero order in the time correlation, in first example the mean of the CTEP equal zero while in other example the mean is nonzero, also we find the variances in the two examples are different. In the two examples we can not obtain on the linear relation be- tween the variance and the mean at any order in time correlation, while [

We thanks Iraqi Ministry of Higher Education and Scientific Research, specifically Iraqi Cultural Relations and Scholarship Department and Cultural Attach in Tehran.

N. J. Hassan,A. Pourdarvish,J. Sadeghi, (2016) The Harmonic Oscillator with Random Damping in Non-Markovian Thermal Bath. World Journal of Mechanics,06,238-248. doi: 10.4236/wjm.2016.68019