# Stochastic Thermodynamics of Brownian Motion

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## Abstract

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## 1. Introduction

## 2. Non-Equilibrium Thermodynamics of Brownian Motion

#### 2.1. Conservation of Mass

#### 2.2. Equation of Motion

#### 2.3. Conservation of Energy and the First Law

#### 2.4. The Second Law

- The dissipation due to the thermodynamic force associated to diffusion, $-{T}^{-1}\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}\xb7\nabla \mu $.
- The dissipation due to the conversion of potential energy into internal energy, ${T}^{-1}\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}\xb7\mathbf{F}$.
- The production of entropy related to the transformation of kinetic energy into internal energy, $-{T}^{-1}\phantom{\rule{0.166667em}{0ex}}\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}\xb7\partial \mathbf{v}/\partial t$.

## 3. Stochastic Thermodynamics: Extended Local Equilibrium Approach

#### 3.1. No External Force

#### 3.2. Overdamping Limit

## 4. Single Particle View

#### 4.1. Time-Independent Potential

#### 4.2. Overdamping Limit

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Modified Bessel function of the second kind corresponding to the distribution (32) where ${\mathsf{\Delta}}_{i}\phantom{\rule{0.166667em}{0ex}}s$ is expressed in units of $n\phantom{\rule{0.166667em}{0ex}}{k}_{B}$.

**Figure 2.**Snapshots of the distribution of the entropy production $T\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Delta}}_{i}s$ obtained from 50,000 realizations of a numerical integration of Equation (22) and substitution in (23). We used a spatially-resolved Euler-Maruyama scheme with a spatial step $dx=0.1$ and a temporal step $dt={10}^{-6}$ in a system of linear size $L=200$. As discussed earlier, the integral is to be taken in the sense of Stratonovich [12]. The parameter values are: $\rho =\beta =1$, ${k}_{B}\phantom{\rule{0.166667em}{0ex}}T=0.01$, $k=2$ and ${\mathrm{v}}^{*}=0.1$. The value of the entropy production has been taken in $x=50$ at $t=1\times {10}^{-4}$ (black histogram), $t=5\times {10}^{-2}$ (red histogram) and $t=0.1$ (blue histogram). The black dashed curve is the modified Bessel function (32), while the red and blue ones are Gaussian fittings. For clarity of presentation, the values of entropy production have been divided by ${k}_{b}\phantom{\rule{0.166667em}{0ex}}\tau $.

**Figure 3.**Distribution of the total work W, the dissipated work ${W}_{diss}$ and the external work ${W}_{ext}$ obtained from 50,000 realizations of a numerical integration of Equation (41) followed by substitution in Equations (42)–(43), respectively. We used an Euler-Maruyama scheme with a temporal step $dt={10}^{-6}$ for a total time $\tau =12$. The parameter values are $\gamma ={k}_{B}\phantom{\rule{0.166667em}{0ex}}T=F=1$. Note that to ensure that initial conditions are sampled over the stationary distribution, a relaxation period of 100 time units has been included before statistics were taken.

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**MDPI and ACS Style**

Nicolis, G.; De Decker, Y.
Stochastic Thermodynamics of Brownian Motion. *Entropy* **2017**, *19*, 434.
https://doi.org/10.3390/e19090434

**AMA Style**

Nicolis G, De Decker Y.
Stochastic Thermodynamics of Brownian Motion. *Entropy*. 2017; 19(9):434.
https://doi.org/10.3390/e19090434

**Chicago/Turabian Style**

Nicolis, Grégoire, and Yannick De Decker.
2017. "Stochastic Thermodynamics of Brownian Motion" *Entropy* 19, no. 9: 434.
https://doi.org/10.3390/e19090434