# Kinetic Energy of a Free Quantum Brownian Particle

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## Abstract

**:**

## 1. Introduction

- The system is at thermal equilibrium of temperature T.
- The state of the system is described by the Gibbs canonical probability distribution.
- The Gibbs probability distribution describes an equilibrium state of the system in the limit of weak coupling with the thermostat.
- The Gibbs probability distribution does not depend on the system-thermostat coupling constant.

## 2. Hamiltonian Model and Generalized Langevin Equation

## 3. Fluctuation-Dissipation Theorem

## 4. Dissipation Function

- -
- In the classical limit, it is a direct relation between the dissipation kernel $\gamma \left(t\right)$ and the correlation function $C\left(t\right)$ : $C\left(t\right)={k}_{B}T\gamma \left(t\right)$. Therefore, for Equation (25), the correlation function is sufficiently universal and has been considered for many systems.
- -
- When $\mathsf{\Omega}=0$, it reduces to the exponential form and is known as a Drude model. Moreover, it has been considered in relation to colored noise problems.
- -
- When $\mathsf{\Omega}=0$ and ${\tau}_{c}\to 0$, then $\gamma \left(t\right)$ tends to the Dirac delta function, and the integral term reduces to ${\gamma}_{0}\dot{x}\left(t\right)$ (no memory effects), which is the well-known Stokes force with ${\gamma}_{0}$ interpreted as a friction (damping) coefficient. Here, the parameter ${\gamma}_{0}$ plays the role of coupling the strength of the Brownian particle to the thermostat.
- -

## 5. Generalized Langevin Equation as a Set of Differential Equations

## 6. Average Kinetic Energy in Equilibrium

## 7. Discussion

#### 7.1. Average Kinetic Energy in Terms of Series

- (i)
- ${E}_{k}$ increases monotonically to infinity when the coupling strength ${\mu}_{0}$ increases to infinity.
- (ii)
- When the decay rate $\epsilon =1/{\tau}_{c}$ grows from zero to infinity, ${E}_{k}$ grows from its classical value to infinity. In other words, for a very long decay time ${\tau}_{c}$, kinetic energy approaches its classical value Equation (4). When ${\tau}_{c}\to 0$, the kinetic energy diverges.
- (iii)
- For $\mathsf{\Omega}=0$ (the Drude model), ${E}_{k}$ starts from some maximal value for a given set of parameters, and next, it decreases when $\mathsf{\Omega}$ increases.

#### 7.2. High Temperature Regime

#### 7.3. Low Temperature Regime

- (i)
- ${E}_{0}$ increases monotonically from zero to infinity when the coupling strength ${\mu}_{0}$ increases from zero to infinity.
- (ii)
- When the decay rate $\epsilon =1/{\tau}_{c}$ grows from zero to infinity, ${E}_{0}$ grows from zero to infinity.
- (iii)
- For $\mathsf{\Omega}=0$ (the Drude model), ${E}_{0}$ starts from some maximal value for a given set of parameters, and next, it decreases when $\mathsf{\Omega}$ increases.

#### 7.4. Regime of Long Memory Time

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Putzer Algorithm

## Appendix B. Calculation of I(ω)

## Appendix C. Series Expansion for Kinetic Energy

## References

- Terletskií, Y.P. Statistical Physics; North-Holland: Amsterdam, The Netherlands, 1971. [Google Scholar]
- Feynman, R.P. Statistical Mechanics; Westview Press: Reading, PA, USA, 1972. [Google Scholar]
- Magalinskij, V.B. Dynamical Model in the Theory of Brownian Motion. J. Exp. Theor. Phys.
**1959**, 36, 1942–1944. [Google Scholar] - Weiss, U. Quantum Dissipative Systems; World Scientific: Singapore, 2008. [Google Scholar]
- Łuczka, J. Non-Markovian stochastic processes: Colored noise. Chaos
**2005**, 15, 026107. [Google Scholar] [CrossRef] [PubMed] - Byron, F.W.; Fuller, R.W. Mathematics of Classical and Quantum Physics; Dover: New York, NY, USA, 1992; Volume 2. [Google Scholar]
- Ullersma, P. An exactly solvable model for Brownian motion: I. Derivation of the Langevin equation. Physica
**1966**, 32, 27–55. [Google Scholar] [CrossRef] - Ford, G.W.; Kac, J. On the quantum Langevin equation. J. Stat. Phys.
**1987**, 46, 803–810. [Google Scholar] [CrossRef] - De Smedt, P.; Dürr, D.; Lebowitz, J.L. Quantum system in contact with a thermal environment: Rigorous treatment of a simple model. Commun. Math. Phys.
**1988**, 120, 195–231. [Google Scholar] [CrossRef] - Van Kampen, N. Derivation of the quantum Langevin equation. J. Mol. Liq.
**1997**, 71, 97–107. [Google Scholar] [CrossRef] - Ford, G.W.; Lewis, J.T.; O’Connell, R.F. Quantum Langevin equation. Phys. Rev. A
**1988**, 37, 4419. [Google Scholar] [CrossRef] - Hänggi, P.; Ingold, G.-L. Fundamental Aspects of Quantum Brownian Motion. Chaos
**2005**, 15, 026105. [Google Scholar] [CrossRef] [PubMed] - Callen, H.B.; Welton, T.A. Irreversibility and Generalized Noise. Phys. Rev.
**1951**, 83, 34. [Google Scholar] [CrossRef] - Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys.
**1966**, 29, 255. [Google Scholar] [CrossRef] - Moler, C.; Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev.
**2003**, 20, 801–836. [Google Scholar] [CrossRef] - Putzer, E.J. Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients. Am. Math. Mon.
**1966**, 73, 2–7. [Google Scholar] [CrossRef] - Hakim, V.; Ambegaokar, V. Quantum theory of a free particle interacting with a linearly dissipative environment. Phys. Rev. A
**1985**, 32, 423. [Google Scholar] [CrossRef] - Grabert, H.; Schramm, P.; Ingold, G.L. Localization and anomalous diffusion of a damped quantum particle. Phys. Rev. Lett.
**1987**, 58, 1285. [Google Scholar] [CrossRef] [PubMed] - Grabert, H.; Schramm, P.; Ingold, G.L. Quantum Brownian motion: The functional integral approach. Phys. Rep.
**1988**, 168, 115–207. [Google Scholar] [CrossRef] - Debnath, L.; Bhatta, D. Integral Transforms and Their Applications; Champman & Hall/CRC: Edinburg, TX, USA, 2008; p. 268. [Google Scholar]
- Ankerhold, J.; Pechukas, P.; Grabert, J. Strong Friction Limit in Quantum Mechanics: The Quantum Smoluchowski Equation. Phys. Rev. Lett.
**2001**, 87, 086802. [Google Scholar] [CrossRef] [PubMed] - Maier, S.A.; Ankerhold, J. Quantum Smoluchowski equation: A systematic study. Phys. Rev. E
**2010**, 81, 021107. [Google Scholar] [CrossRef] [PubMed] - Łuczka, J.; Rudnicki, R.; Hänggi, P. The diffusion in the quantum Smoluchowski equation. Phys. A
**2005**, 351, 60–68. [Google Scholar] [CrossRef] - Machura, Ł.; Kostur, M.; Talkner, P.; Łuczka, J.; Hänggi, P. Quantum diffusion in biased washboard potentials: Strong friction limit. Phys. Rev. E
**2006**, 73, 031105. [Google Scholar] [CrossRef] [PubMed] - Machura, Ł.; Kostur, M.; Hänggi, P.; Talkner, P.; Łuczka, J. Consistent description of quantum Brownian motors operating at strong friction. Phys. Rev. E
**2004**, 70, 031107. [Google Scholar] [CrossRef] [PubMed] - Bhadra, C.; Banerjee, D. System-reservoir theory with anharmonic baths: A perturbative approach. J. Stat. Mech.
**2016**, 4, 043404. [Google Scholar] [CrossRef] - Carlaesso, M.; Bassi, A. Adjoint master equation for quantum Brownian motion. Phys. Rev. A
**2017**, 95, 052119. [Google Scholar] [CrossRef]

**Figure 1.**Average kinetic energy of the free Brownian particle as a function of rescaled temperature. (

**a**) The influence of the rescaled particle-thermostat coupling strength ${\tilde{\mu}}_{0}={\mu}_{0}/\epsilon $. The rescaled energy is $\tilde{E}={E}_{k}/\hslash \epsilon $, and the rescaled temperature is $\tilde{T}={k}_{B}T/\hslash \epsilon $. The rescaled $\tilde{\mathsf{\Omega}}=\mathsf{\Omega}/\epsilon =1$. (

**b**) The influence of the rescaled inverse decay time $\tilde{\epsilon}=\epsilon /{\mu}_{0}$. The rescaled energy is $\tilde{E}={E}_{k}/\hslash {\mu}_{0}$, and the rescaled temperature is $\tilde{T}={k}_{B}T/\hslash {\mu}_{0}$. The rescaled $\tilde{\mathsf{\Omega}}=\mathsf{\Omega}/{\mu}_{0}=1$. (

**c**) The influence of the rescaled frequency $\tilde{\mathsf{\Omega}}=\mathsf{\Omega}/{\mu}_{0}$. The rescaled energy is $\tilde{E}={E}_{k}/\hslash {\mu}_{0}$, and the rescaled temperature is $\tilde{T}={k}_{B}T/\hslash {\mu}_{0}$. The rescaled $\tilde{\epsilon}=\epsilon /{\mu}_{0}=1$.

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Bialas, P.; Łuczka, J.
Kinetic Energy of a Free Quantum Brownian Particle. *Entropy* **2018**, *20*, 123.
https://doi.org/10.3390/e20020123

**AMA Style**

Bialas P, Łuczka J.
Kinetic Energy of a Free Quantum Brownian Particle. *Entropy*. 2018; 20(2):123.
https://doi.org/10.3390/e20020123

**Chicago/Turabian Style**

Bialas, Paweł, and Jerzy Łuczka.
2018. "Kinetic Energy of a Free Quantum Brownian Particle" *Entropy* 20, no. 2: 123.
https://doi.org/10.3390/e20020123