Received 10 July 2016; accepted 14 August 2016; published 17 August 2016
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  -  respectively. The effect of correlations in the random driving force on the stationary probability density and its moments were studied  -  . The exact formula is found for the first moment and the system of equations for second moments for harmonic oscillator with random mass  . The analytical expressions are derived for the sta- tionary probability density of the particle’s energy  . Both theoretical approaches were formulated in the context of the standard Langevin equation   , where the friction force was proportional to the speed with a constant friction coefficient and additive Gaussian white noise. The non-equilibrium process efforts are com- monly formulated in the form of stochastic thermodynamic culminates into fluctuation relations connecting exten- sive thermodynamic variables such as work, free energy, and entropy  -  . The violation of the Markovian approximation of the environment leads to generation of additional entropy  . The ideas behind the entropy production are studied and some insights are given about relevance  . The statistical properties of stochastic entropy production associated with the non stationary transport of heat through system coupled to a time depen- dent nanisothermal heat bath  . When the harmonic oscillator with external noise in non-Markovian thermal bath, the cumulants of order two and three contain the natural effects of the non-Markovian bath through the noise correlation time, consequently the non Gaussian characteristic of the totel entropy change drives to a breakdown of the usual fluctuation theorems  . The purpose of this paper is discussing the change in total entropy production for the harmonic oscillator with random damping in non-Markovian thermal bath and stu- dying the fluctuation theorem (FT) at any order in time correlation when the harmonic potentials are represented the time dependent driving force or the time dependent dragging force where this force is arbitrary time depen- dent. We derive in this paper the stochastic differential equation (SDE) driving by the multiplicative colored noise and translate it into additive colored noise by changing variables from x to y. In this side, in order to compatible the additive colored noise system with potential, we change variables in harmonic potentials (time dependent driving force and time dependent dragging force). We calculate the mean, variance and the distri- bution function for the change in total entropy production in new formulas of the harmonic potentials. We show that in our model, the fluctuation theorem is invalid at any order in the noise correlation time. Finally, we pre- sent our conclusion and we give the future works. This paper can be divided by six sections. In Section 2, we define the new model based on the SDE driving by additive colored noise in the overdamped approximation, and we find the Fokker Planck equation which it is associated of the SDE. In Section 3, we change variables in Equation (1) from x into y, this represent first example. In this example, we find the change in total entropy, mean and variance. In Section 4, we change variables in Equation (2), this represent second example. In this example, we also compute the change in total entropy, mean and variance. In Section 5, we show that the FT is invalid at any order in the time correlation. Finally, we introduce the conclusions and future works.
2. Stochastic Differential Equation (SDE) Driving by Additive Colored Noise in Overdamed Approximation
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 is Ornstein-Uhlenbeck noise (special type of colored noise), is friction constant, is corre- lation time, is the harmonic potential, is arbitrary time depend force, is particle’s position and is the velocity. The is Gaussian distribution with zero mean and correlation function is,
where such that is Boltzmann constant and T is heat temperature. We assume that the following ,
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 and, the Equation (7) read as,
By using power series at first order in the noise, the above equation become,
let and where. Equation (9) is SDE driving by multipli-
cative colored noise. To translate Equation (9) from multiplicative colored noise into additive ,we must divided Equation (9) by, one can obtain,
then the translation  is defined as,
then the Equation (10) is,
Equation (13) represent SDE driving by additive colored noise. The Fokker Planck equation  is defined,
where the initial condition, we solve the Fokker Planck equation by Fourier trans- formation  as,
we take the time derivative into above equation, we have,
Assume that, then we have,
where then the transition probability is,
To obtain the initial distribution function we must assume that at zero order in time correlation that
mean (converge to zero), then we get,
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 have same exponential distribution.
3. The First Example
In this section, we change variables in the time dependent driving force from x into y which is defined as
where is arbitrary time depend force and under the new formula of the harmonic potential, we calculate the change in the total entropy production (CTEP), mean and the variance. From Equation (12), we have,
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. The based on the stochastic thermodynamic approach    , the first law thermodynamic like can be defined as,
where the work can be computed  as,
The mean of the work is calculated as,
where the quantity can be found as,
Putting Equation (28) inside Equation (27), we get,
the variance of the work can be calculated as,
where the quantity is found as,
Substituting Equation (31) inside Equation (30). one can obtain,
The change in the environment entropy is obtain as,
The change in entropy of the system is given as,
Now, we can find the CTEP as,
where and the mean of the is,
we must calculate the following quantities, and as,
note here. Putting Equation (29) and an above quantities’s values inside Equation (36), we get,
before we find the variance of the CTEP, we make some the following assumptions, let with it coefficient, with it coefficient, with it coefficient, with it coefficient and with it coefficient,
we must calculate the following quantities, ,
, and as,
Putting Equation (32) and an above quantities’s values in Equation (41), one can obtain,
At zero order in time correlation, the change in totel entropy production in here read as,
where then the mean of is,
where and. Here we note that: First, the and are exponential distributions but they in  are Gaussian. The second, at zero order in time correlation we not found linear relation between the mean and variance of the change in totel entropy , while it exist in  .
4. The Second Example
In this section , we change variables in the time dependent driving force from x into y which is defined as
where is arbitrary time depend force and under the new formula of the harmonic potential , we calculate the change in the total entropy production (CTEP), mean and the variance. Substituting Equation (22) in Equation (51), one can obtain,
Equation (52) represent second new formula of harmonic potential in y .
where we assume that. The work can be computed as,
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 is defined as,
where. Note that, , that mean of the change in entropy of the environ- ment different in two examples. The mean of the change in entropy of the environment is calculated as,
The variance of the is,
since the variance of the is,
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 as,
The mean of the is,
The variance of the is,
At zero order in time correlation , the CTEP is,
the mean is,
and variance is,
where and. Here note that: At zero order in time correlation, in first example
while in second example, also the and in two exam-
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.  is shown that the entropy variance is same in his two examples at first order in time correlation and it found the relation between the mean and variance of the CTEP at first order in time correlation.
5. Fluctuation Theorem
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 is generating function and is cumulant function, such that,
, ... are first cumulant, second cumulant, third cumulant and so on. We calculate the distribution function of the CTEP in two perspectives. The first perspective, at second order of approximation in u, the Equation (62) becomes,
from above equation, we find that the is generating function of the Gaussian distribution function, that mean has Gaussian distribution with mean and variance. The FT is invalid as
The other perspective, at any order in u, this perspective is studied in  , it show that the distribution function of the CTEP is non Gaussian and the FT is invalid as
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  becomes valid.
6. Conclusion and Future Work
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  it obtained this relation at zero order in the time correlation. The FT in our work is invalid at any order in the time correlation while in  the FT is valid at zero order in the time correlation. Finally , we will study the harmonic oscillator with random frequency in Markovian and non-Markovian thermal bath. This problem will be done in future.
We thanks Iraqi Ministry of Higher Education and Scientific Research, specifically Iraqi Cultural Relations and Scholarship Department and Cultural Attach in Tehran.
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