A Path integral solution of the Fokker-Planck equation

Work distribution function for a Brownian particle driven by a nonconservative force

Abstract

We derive the distribution function of work performed by a harmonic force acting on a uniformly dragged Brownian particle subjected to a rotational torque. Following the Onsager and Machlup’s functional integral approach, we obtain the transition probability of finding the Brownian particle at a particular position at time given that it started the journey from a specific location at an earlier time. The difference between the forward and the time-reversed form of the generalized Onsager-Machlup’s Lagrangian is identified as the rate of medium entropy production which further helps us develop the stochastic thermodynamics formalism for our model. The probability distribution for the work done by the harmonic trap is evaluated for an equilibrium initial condition. Although this distribution has a Gaussian form, it is found that the distribution does not satisfy the conventional work fluctuation theorem.

1 Introduction

Fluctuations are ubiquitous in most of the physical processes we observe in nature. In general, though the effect of fluctuations could be ignored for the coarse grained description of the average thermodynamic quantities, these are no longer negligible for small systems, like, biomolecules, nano-machines etc. For instance, the movement of a molecular motor of nanometer size is affected significantly by the mean thermal energy of the surrounding medium which causes the motor’s position to fluctuate. In linear irreversible processes, these fluctuations obey a general principle known as the fluctuation-dissipation theorem (1); (2). However, in the case of systems far away from equilibrium, the assumptions of the linear-response theory are not valid since the fluctuations of the thermodynamic variables do not follow the linear law of relaxation.

Since the finding of the fluctuation relations for the probabilities of observing the entropy producing and consuming trajectories in simulations of shearing fluid nearly two decades ago (3), there has been a vast number of studies (4); (5); (6); (8); (9); (10); (17); (18); (19); (11); (12); (13); (14); (15); (16); (7) extending the ideas and formalisms of linear irreversible thermodynamics to nonequilibrium systems. The emphasis of the fluctuation relations has been on quantifying the asymmetry between the probability distributions of a time-integrated quantity to have values in the forward and time-reversed processes, and it is expressed in the mathematical form as . could represent the work done by an external force, dissipated heat, entropy production etc, and stands for the same thermodynamic quantity in the time-reversed process. These relations, valid for systems far-from-equilibrium, could be reduced to the fluctuation-dissipation theorem for near-equilibrium processes (7). Of the many ramifications of the fluctuation relations, the most notable results are the Jarzynski equality (8) and Crooks work relation (9) which relate the asymmetry to the free energy difference between the initial and final equilibrium states which are connected through a reversible or irreversible transformation. Although, to prove the fluctuation relations, one does not always require to obtain the detailed distribution functions, to verify them experimentally, however, it is preferable to get the histograms of the relevant thermodynamic quantities. In many situations (20), a detailed knowledge of the distribution of the nonequilibrium work might be essential to compute the equilibrium free energy difference utilizing the Jarzynski equality. Despite the significant progress in the theoretical studies of these distribution functions for work, heat, entropy production etc for a variety of nonequilibrium systems (17); (18); (19); (21); (22); (23); (24); (20); (25); (28); (29); (26); (27) (and the references therein), the derivation of the exact analytical expressions for these distributions are not so many. This is possibly due to the increasing difficulty in obtaining the analytical solutions in closed forms as the dynamics of the system becomes more complex.

In this paper, we derive the positional probability distribution and the work distribution function for a Brownian particle using the Onsager-Machlup’s functional integral approach. In their theory of linear relaxation processes, Onsager and Machlup developed a path integral approach (30); (31) for the probability distribution functions of trajectories in the configuration space with specific initial and final states. Using a variational technique they subsequently showed that this approach leads to Onsager’s minimal energy dissipation. Recently, there have been many efforts to generalize the Onsager-Machlup theory for systems in nonequilibrium steady states (32); (33); (34); (35); (36); (37); (38); (39). Extending this approach, Taniguchi and Cohen (36); (37) obtained the distribution functions of work, dissipated heat for a uniformly dragged Brownian particle. Further, after expressing the Onsager-Machlup Lagrangian in terms of the dissipation functions and the entropy production rate, they showed how these fluctuating quantities satisfy the energy conservation law and the second law of thermodynamics. Similar variational scheme has also been considered to obtain the probability distributions of thermodynamic quantities for a variety of systems (40).

In the present work, we consider two dimensional motion of a Brownian particle which is subjected to a nonconservative force and a confining harmonic potential with its minimum moving with a uniform velocity . The nonconservative force

(1)

with denoting the strength of the force, produces an anticlockwise drift about the origin. The nonconservative force can arise due to a vortex centered at the origin of the system in the surrounding liquid (41). Both the time dependent confining potential and the nonconservative force are responsible for breaking the detailed balance and driving the system out of equilibrium. For this model, we derive the joint probability distribution which gives the probability of finding a Brownian particle at a given position at a specific time. After defining the forward and the time-reversed trajectories, we obtain the expression for the dissipated heat and the energy conservation relation. In the process, we identify various works, namely, the work done by the harmonic trap which can be identified as the thermodynamic work (8) and the mechanical work done by the nonconservative force (28); (29). Finally, using a variational approach, we obtain the distribution function for the work done by the trap. The work distribution has a Gaussian form with the mean and the variance having a nontrivial oscillatory dependence originating from the nonconservative force. Although, this distribution is Gaussian, it does not satisfy the conventional fluctuation relation. Physically, the presence of the nonconservative force makes the role of the thermal fluctuation negligible. Similar deviation can also be found in (42); (43) for the motion of a torsion pendulum driven by an external time dependent torque.

The rest of the paper is organized in the following way. In section 2, we introduce two coupled Langevin equations describing the motion of the Brownian particle in the plane. In section 3, we find out the joint probability distribution for the position of the Brownian particle. Starting with the Fokker-Planck equation for the joint probability distribution, we express first the transition probability in a functional integral form and subsequently evaluate the path integrals employing Onsager-Machlup’s variational approach. The joint probability distribution is obtained from the expression of the transition probability. From the generalized Onsager-Machlup Lagrangian, we define in section 4 the thermodynamic work, mechanical work, dissipated heat and the entropy production of the medium. We further show here the validity of the energy conservation relation and the second law of thermodynamics for a trajectory. In section 5, we obtain the explicit form of the distribution function of the thermodynamic work. We summarize our findings and discuss possible future research on this model in section 6. Some of the details of the derivations and algebra are presented in the appendices.

2 The model

The Brownian particle considered here moves in a plane under the influence of the potential and a nonconservative force as given in equation (1) (41); (29). Here is the constant velocity with which the harmonic trap moves along the direction.

The overdamped Langevin equations describing the motion of the particle in the - plane are expressed as

(2)
(3)

where, is the viscous drag coefficient, and is the relaxation time of the trap. The overdots imply derivatives with respect to time, . and represent the Gaussian-distributed thermal noise having zero mean and correlations as

(4)

Here and correspond to the coordinates.

3 Probability distribution functions related to the location of the particle

The joint probability distribution is defined as the probability of finding the Brownian particle at position at time . The time evolution of the joint probability distribution is governed by the Fokker-Planck equation which can be written as

(5)

where, the Fokker-Planck operator is given by,

(6)

with the diffusion constant, . Let us consider the simplest situation where . Writing the Fokker-Planck equation in terms of the probability current, , we have

(7)

where

(8)

Substituting (with ), we obtain a vanishing stationary current corresponding to the equilibrium situation of a fixed harmonic potential. This is the situation where the detailed balance is satisfied. The nonconservative force which cannot be expressed in the form of a gradient of a potential violates the detailed balance.

can be expressed in terms of the transition probability which gives the probability of finding a Brownian particle at position at time , given that it was at at an initial time . For simplicity, we choose and denote the transition probability as . The Fokker-Planck equation (5) allows us to obtain the following path integral representation for the transition probability (44) (some of the details of the derivation are presented in appendix-A)

(9)

is the generalized Onsager-Machlup Lagrangian having the form

(10)

We evaluate the functional integrals (9) using a variational approach (30); (31); (45); (36); (37), in which one evaluates the most probable path that gives the most significant contribution to the functional integral. This is expected to be a good approximation for low noise strength or (45). The approximate form of the transition probability can be written as

(11)

where the object in the square bracket is evaluated over the most probable path leading to the largest value of the exponential. The most-probable path is determined by using the extremization condition

(12)

Doing a Taylor-expansion around the optimal values, , and retaining fluctuations about the optimal path, and up to second order the transition probability in (9) can be written as,

(13)

Since we consider trajectories with fixed end-points and allow no variation at the end points, and satisfy Dirichlet boundary conditions . As a consequence of this, the fluctuation factor

(14)

is a function of the final time only. Instead of evaluating the path integrals in Eq. (14) directly, we determine using the normalization condition of the transition probability, i.e., the total probability of finding the Brownian particle at any final value of after time is unity. It can be proved, a posteriori, from the normalization property of , that the final expression of has the form

(15)

The extremization condition (12) leads to two coupled Euler-Lagrange equations for the optimal paths and

(16)
(17)

The solutions of the above equations are

(18)
(19)

where . The constants and are calculated using the boundary conditions . These constants are

(20)
(21)
(22)
(23)

Substituting Eqs.(14), (18) and (19) into Eq.(13), and performing the integration over time, we have the complete expressions for as

(24)
Figure 1: Plot of the joint distribution function for the parameter values and at final time .

To obtain the joint probability distribution , we integrate the above expression over the initial points, assuming that the system was initially in thermal equilibrium with the joint probability distribution
. So,

(25)

where

(26)
(27)

During simplifications, we have used the relation, . For a particular set of parameter values, the joint probability density is plotted in Fig.(1).

The marginal probability is defined as the probability of finding the particle at a given or for arbitrary values of or , respectively. After integrating with respect to the appropriate end point variable, the marginal probability distributions are found as

(28)

and

(29)

Eqs. (25), (28) and (29) are the main results of this section.

4 Generalized Onsager-Machlup Lagrangian and stochastic thermodynamic quantities

In this section, we establish an energy-balance relation and a second law type formulation valid for a single trajectory. We consider that, for every forward trajectory specifying the system’s evolution over time interval , there exists a time-reversed trajectory over the same interval. The probability measure of the forward path could be expressed in terms of the transition probability up to a normalization factor as,

(30)

with the generalized Onsager-Machlup Lagrangian as given in Eq.(10). Similarly, the transition probability corresponding to the time reversed trajectory is expressed as (36); (37)

(31)

where is obtained from under the time-reversal changes and .

4.1 Entropy production in the medium

The ratio of the transition probability for the forward trajectory to that of the reversed trajectory is given by

(32)

where

(33)

The entropy production in the medium satisfies the detailed fluctuation theorem (28)

(34)

From relation (32) and (33), we have

(35)

where is the change in the potential energy and the explicit form of the second term is

(36)

4.2 Work and dissipated heat

Since the heat transferred to the medium is , we use Eq. (35) to find

(37)

In the last equality, the dissipated heat is expressed in terms of the change in the potential energy and the work that consists of two parts, namely, the thermodynamic work, , which is the work done on the system by the harmonic trap (46); (21); (18); (19) and the mechanical work, . Similar partition of the total work into physically meaningful works can be found in recent literatures (47); (42); (48). For further discussions on various definitions of work and their behavior under gauge transformation, readers are referred to (48). Together with these definitions, Eq.(37) becomes the statement of the energy conservation principle valid on each trajectory (13); (14); (46).

Next, to show the second law type formulation for the medium entropy production in the framework of the Onsager-Machlup theory (36); (37), we split the Lagrangian into time reversal symmetric, , and anti-symmetric, , components and arrive at

(38)

Using Eqs. (32) and (34), in the above equation has been replaced by the rate of entropy production . It is evident from Eq.(10) that the average path described by

(39)
(40)

corresponds to the minimum value of the integral . Substituting (39) and (40) into the relation

(41)

we find

(42)

Eq. (42), ensuring , shows the consistency with the second law of thermodynamics.

5 Distribution of the work performed by the moving trap

In this section, we derive the distribution of the work done by the harmonic trap on the Brownian particle. The work done or the energy put into the system by the trapping potential over time is

(43)

In order to obtain the distribution of this work, we need to perform functional averages of this quantity over all possible paths as well as integrals over the initial and final points (24); (36); (37) of the path. The probability measure of such a path is given by,

(44)

So, the probability distribution of the work performed by the trap, henceforth denoted as , is

(45)

where

(46)

with

(47)

In order to derive the distribution function at a finite time, we consider the system to have evolved from the initial equilibrium distribution, .

To evaluate the functional integrals in (46), we follow the variational approach discussed earlier in section 3. We seek the most probable trajectory , which corresponds to the largest contribution to the path integral. The extremization condition, , leads to two Euler-Lagrange equations

(48)
(49)

which are supplemented with the boundary conditions, and . After some simplifications, the two Euler-Lagrange equations can be expressed in terms of higher derivatives of a single variable only. These are

(50)
(51)

where, . The optimal solutions are

(52)
(53)

The explicit dependence of the constants and on the boundary conditions and other parameters of the problem is shown in appendix B. Incorporating the quadratic term arising from the fluctuations around the most probable path, we write Eq.(46) as,

(54)

Substituting the solutions (52) and (53) into (54) and subsequently performing first the integral over time and then over final and initial positions (some of the intermediate calculations are shown in appendix B), we have

(55)

where

(56)
(57)
Figure 2: Plot of the symmetry function with for the parameter values and , at the final time . The dashed-dot line corresponds to the situation where the symmetry function follows the fluctuation theorem. The deviation from the fluctuation relation increases for larger values of .

Substituting (55) into Eq.(45) and doing the final Gaussian integral over , we find the final form of the work distribution as

(58)

The known result of the work distribution function for a uniformly dragged colloidal particle (21); (18); (19); (22); (23) can be retrieved by substituting in our general result. The work distribution function, in this case, obeys the fluctuation theorem. The symmetry function defined as , therefore, has a value