# Fokker-Planck Equation and Thermodynamic System Analysis

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Non-equilibrium Fokker-Planck Equation

_{i}is the position of the i-th particle,

**x**= {x

_{i}}, f

_{i}is the force acting on the i-th particle, r

_{i}is the noise-induced drift term due to diffusion gradient effects (if the noise is state dependent). The noise term r

_{i}is a stochastic variable that obeys a Gaussian probability distribution with correlation function such as:

_{i}≥ 0 is a different constants for each particle. This implies the Onsager reciprocity relation δ

_{ij}= δ

_{ji}for the dumping coefficients. The δ-function of the correlations in time means that r

_{i}(t) at a timer t is assumed to be completely uncorrelated with it a any other time.

**x**,t), usually named Smoluchowski equation, can be written as [8,14]:

**x**= {x

_{i}}, under boundary conditions related to the behavior of P(

**x**,t) and J

_{i}(

**x**,t) at the boundary surface of the integration space itself. The condition of irreversibility can be expressed as follows:

_{es}is the work done by the environment (the world outside the system) on the system, in other words the work done from external forces on the border of the system, W

_{fe}is the work lost due to external irreversibility (outside energy dissipation), ΔE

_{k}is the kinetic energy of the system, W

_{i}is the internal work, such that:

_{fi}the work lost due to internal irreversibility. Moreover, the following relation must be taken in account [15,16]:

_{se}is the work done by the system on the environment, i.e., the work done from internal forces on the border of the system. It is then possible to obtain the following two equivalent equations from the First Law of Thermodynamics:

_{λ}. These results underline the connection between the entropy generation and the work (energy) inside the system balanced on the boundary between the system and the environment and even provide a new approach to highlight the interaction between the system and its environment. Indeed, we are highlighting that irreversibility is the core of the whole entropy generation approach. The existence of irreversibility is due to the work lost, and this could be considered as a “natural” way of communication between the complex system and its environment. One important aspect is that, being the complex system accessible just due to the irreversible processes, the energy lost towards the environment can be seen as the information lost from the system, but this information is gained by the environment. Complex systems are all the time in a non-equilibrium situation with a continuous energy transduction: cell molecular machines are typical examples. A cell molecular machine can transport matter along the cell as done by kinesin protein in eukaryotic cells, or can propel the cell through the extracellular media, as the case of bacterial flagellar motor. In both cases they use energy from electrochemical potential variations or from the hydrolysis of the ATP. The ATP diffusion process inside the cell is a diffusion process and it can be worthy studied by Fokker-Planck equation and entropy generation approach as we show in the next paragraph.

## 3. Entropy Generation and Fokker-Planck Equation

_{I}= −Σ

_{γ}p

_{γ}lnp

_{γ}, with respect to p

_{γ}on the path γ. The information entropy is then seen as the logarithm of the number of the outcomes i with non-negligible probability p

_{i}, while in non-equilibrium statistical mechanics approach information entropy is the logarithm of the number of microscopic phase-space paths γ having non-negligible probability p

_{γ}. Following the suggestions of [17] we have to evaluate the most probable macroscopic path realised by the greater number of microscopic paths compatible with the imposed constrained, in full analogy with the Boltzmann microstate counting that claims paths, rather than states, are the central objects of interest in non-equilibrium systems. This because of the presence of non-zero macroscopic fluxes whose statistical description requires to take into account the microscopic behaviour over time. The last statement implies that the macroscopic behaviour is reproducible under known constraints and it is characteristic of each of the great number of microscopic paths compatible with those constraints [17]. From these considerations, it has been proven that the information entropy for the open systems is related to their entropy generation as [18]:

_{γ}= P

_{γ}(

**x**,t). One possible interpretation of (16) is the missing information necessary for predicting which path a system of the ensemble takes during the transition from one state to another. Now, considering Gouy-Stodola theorem [7] the entropy generation can be related to the power lost ${\dot{W}}_{\lambda}$ due to irreversibility as expressed in the relations (14) and (15), and it follows:

_{0}reference temperature (usually environmental temperature is taken), thought as a constant. Now the power lost for irreversibility must be obtained. Starting from the definition [8,14]:

_{ii}= ∂f

_{i}/∂x

_{i}. Using this relation the entropy generation results:

_{0}) and the characteristic time (τ) of the process considered.

**x**represent all the possible microscopic phase-space paths and P

_{γ}(

**x**,t) the associated probability.

## 4. Application to Biological Molecular Machines

_{in}(ϑ) is the internal torque profile generated by chemical processes, τ

_{ext}is the conservative external torque applied on the motor and r(t) is the thermal noise described, for instance, as a Gaussian white noise of zero mean value. This approach can be used for a linear motor such as kinesin protein behaviour in eukaryotic cells. In kinesin the linear length of the step is fixed by the periodicity of the dimers of the microtubule track, while for rotatory motors the angular step depends on the radial symmetry of the motor. Moreover, two different energetic states must be considered:

- the relaxed state, in which the motor does not advance and waits for an energetic input;
- the excited state, in which the energy is transduced producing the power stroke of the motor.

_{λ}can be written as:

_{ATP}is the free energy variation (free energy = ΔG

_{ATP}– W

_{λ}, W

_{λ}is the work lost for irreversibility and is a constant since a specific process is considered) due to the hydrolysis of a single ATP molecule (~21 k

_{B}T = 50 kJ mol

^{−1}, being k

_{B}the Boltzmann constant and T the temperature). The Equation (25) shows the connection to efficiency η of the molecular machine and the probability distribution. We can argue that this probability distribution, is strongly related to molecular machines operations and would be different from normal and ill cell, being different the behaviour of the cells and their interactions with the environment. This equation allows us to link the macroscopic power of the molecular machine with the microscopic chemical reaction probability.

## 5. Conclusions

**PACS Codes:**05.70.-a; 05.10.Gg; 05.70.Ln; 82.40.Bl

## Author Contributions

## Conflicts of Interest

## References

- Jordan, R.; Kinderlehrer, D.; Otto, F. The Variational Formulation of the Fokker-Planck Equation. SIAM J. Math. Anal.
**1998**, 29, 1–17. [Google Scholar] - Gardiner, C.W. Handbook of Stochastic Methods, 2nd ed; Springer: Berlin, Germany, 1985. [Google Scholar]
- Risken, H. The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed; Springer: Berlin, Germany, 1989. [Google Scholar]
- Schuss, H. Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics. SIAM Rev
**1980**, 22, 119–155. [Google Scholar] - Lucia, U. Carnot Efficiency: Why? Physica A
**2013**, 392, 3513–3517. [Google Scholar] - Lavenda, B.H. Thermodynamics of Irreversible Processes; Dover: Mineola, NY, USA, 1993. [Google Scholar]
- Bejan, A. Advanced Engineering Thermodynamics; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- Lucia, U. Thermodynamic Paths and Stochastic Order in Open Systems. Physica A
**2013**, 392, 3912–3919. [Google Scholar] - Tomé, T. Entropy Production in Non-Equilibrium Systems Described by a Fokker-Planck Equation. Braz. J. Phys.
**2006**, 36, 1285–1289. [Google Scholar] - Dewar, R. Information Theory Explanation of the Fluctuation Theorem, Maximum Entropy Production and Self-Organized Criticality in Non-Equilibrium Stationary States. J. Phys. A
**2003**, 36, 631–641. [Google Scholar] - Annila, A. All in Action. Entropy
**2010**, 12, 2333–2358. [Google Scholar] - Wang, Q. Maximum Path Information and the Principle of Least Action for Chaotic System. Chaos Solitons Fractals
**2004**, 23, 1253–1258. [Google Scholar] - Wang, Q. Non Quantum Uncertainty Relations of Stochastic Dynamics. Chaos Solitons Fractals
**2005**, 26, 1045–1053. [Google Scholar] - Lucia, U. Entropy Generation and Fokker-Planck Equation. Physica A
**2014**, 393, 256–260. [Google Scholar] - Wang, Q. Maximum Entropy Change and Least Action Principle for non-Equilibrium Systems. Astrophys. Space Sci.
**2006**, 305, 273–279. [Google Scholar] - Wang, Q. Probability Distribution and Entropy as a Measure of Uncertainty. J. Phys. A
**2008**, 41, 065004. [Google Scholar] - Lucia, U. Irreversible Entropy Variation and the Problem of the Trend to Equilibrium. Physica A
**2007**, 376, 289–292. [Google Scholar] - Lucia, U. Thermodynamic Paths and Stochastic Order in Open Systems. Physica A
**2013**, 392, 3912–3919. [Google Scholar] - Perez-Carrasco, R.; Sancho, J.M. Fokker-Planck Approach to Molecular Motors. Eur. Phys. Lett.
**2010**, 91, 60001. [Google Scholar] - Sharma, V.; Annila, A. Natural Process–Natural Selection. Biophys. Chem.
**2007**, 127, 123–128. [Google Scholar] - Sharma, V.; Kaila, V.R.I.; Annila, A. A Protein Folding as an Evolutionary Process. Physica A
**2009**, 388, 851–862. [Google Scholar] - Annila, A.; Salthe, S. Physical Foundations of Evolutionary Theory. J. Non-Equilib. Thermodyn.
**2010**, 35, 301–321. [Google Scholar] - Annila, A.; Salthe, S. Cultural Naturalism. Entropy
**2010**, 12, 1325–134. [Google Scholar] - Martyushev, L.M. Entropy and Entropy Production: Old Misconceptions and New Breakthroughs. Entropy
**2013**, 15, 1152–1170. [Google Scholar] - Polettini, M. Fact-Checking Ziegler’s Maximum Entropy Production Principle beyond the Linear Regime and towards Steady States. Entropy
**2013**, 15, 2570–2584. [Google Scholar] - Lebowitz, J.L.; Boltzmann’s, Entropy. Large Deviation Lyapunov Functionals for Closed and Open Macroscopic Systems
**2011**, arXiv, 1112.1667.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lucia, U.; Gervino, G.
Fokker-Planck Equation and Thermodynamic System Analysis. *Entropy* **2015**, *17*, 763-771.
https://doi.org/10.3390/e17020763

**AMA Style**

Lucia U, Gervino G.
Fokker-Planck Equation and Thermodynamic System Analysis. *Entropy*. 2015; 17(2):763-771.
https://doi.org/10.3390/e17020763

**Chicago/Turabian Style**

Lucia, Umberto, and Gianpiero Gervino.
2015. "Fokker-Planck Equation and Thermodynamic System Analysis" *Entropy* 17, no. 2: 763-771.
https://doi.org/10.3390/e17020763