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Exact Work Distribution and Jarzynski's Equality of a Relativistic Particle in an Expanding Piston

Xianghang Zhang, Tingzhang Shi, H. T. Quan

TL;DR

This work analyzes non-equilibrium work in a relativistic, one-dimensional piston model containing an ideal gas. The authors derive the exact single-particle trajectory and the full work distribution, and verify Jarzynski's equality in this relativistic setting, noting that the distribution loses zeros and remains finite in the number of particle collisions due to the speed-of-light bound. They demonstrate that in the nonrelativistic limit the results reduce to known Newtonian expressions, while in the relativistic regime significant deviations appear at high temperature and fast piston speeds. Despite these insights, the authors conclude that detecting relativistic effects in practical experiments with current techniques is difficult, underscoring the robustness of Jarzynski's equality across dynamics.

Abstract

We study the non-equilibrium work in a pedagogical model of relativistic ideal gas. We obtain the exact work distribution and verify the Jarzynski's equality. In the non-relativistic limit, our results recover the non-relativistic results [arXiv:cond-mat/0502434]. We also find that, unlike the non-relativistic case, the work distribution no longer has zeros and the number of collisions in this relativistic gas model is finite. In addition, based on an analysis of the experimental parameters, we conclude that it is difficult to detect the relativistic effects of the work distribution of the ideal gas in a piston system with the current experimental techniques.

Exact Work Distribution and Jarzynski's Equality of a Relativistic Particle in an Expanding Piston

TL;DR

This work analyzes non-equilibrium work in a relativistic, one-dimensional piston model containing an ideal gas. The authors derive the exact single-particle trajectory and the full work distribution, and verify Jarzynski's equality in this relativistic setting, noting that the distribution loses zeros and remains finite in the number of particle collisions due to the speed-of-light bound. They demonstrate that in the nonrelativistic limit the results reduce to known Newtonian expressions, while in the relativistic regime significant deviations appear at high temperature and fast piston speeds. Despite these insights, the authors conclude that detecting relativistic effects in practical experiments with current techniques is difficult, underscoring the robustness of Jarzynski's equality across dynamics.

Abstract

We study the non-equilibrium work in a pedagogical model of relativistic ideal gas. We obtain the exact work distribution and verify the Jarzynski's equality. In the non-relativistic limit, our results recover the non-relativistic results [arXiv:cond-mat/0502434]. We also find that, unlike the non-relativistic case, the work distribution no longer has zeros and the number of collisions in this relativistic gas model is finite. In addition, based on an analysis of the experimental parameters, we conclude that it is difficult to detect the relativistic effects of the work distribution of the ideal gas in a piston system with the current experimental techniques.
Paper Structure (12 sections, 58 equations, 7 figures)

This paper contains 12 sections, 58 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Relativistic Piston Model
  3. Trajectory of a single particle
  4. Direct verification of Jarzynski's equality in the expanding relativistic piston model
  5. Distribution of work
  6. Nonrelativistic Limit
  7. Limit of sast-moving piston
  8. Conclusion
  9. Covariant description of the trajectory
  10. Jarzynski's equality under the framework of relativistic mechanics
  11. Derivation of Liouville's Theorem in the Relativistic Piston Model
  12. Details of the Integration

Figures (7)

  • Figure 1: Relativistic and nonrelativistic work distribution with different parameters. We use hydrogen atoms as an example and suppose that the length of the cylinder is $L= 1~\mathrm{cm}$ (which is much larger than the thermal length of the atoms). The parameters are chosen to be (a) $\tau=0.3~\mathrm{ns}, v_p=3\times 10^7~\mathrm{m/s}, T=3\times 10^{12}~\mathrm{K}$, and $\beta = 3$; (b) $\tau=3 ~\mathrm{ns}, v_p=3\times 10^6~ \mathrm{m/s}, T=1\times 10^{12}~\mathrm{K}$, and $\beta = 10$; and (c) $\tau=30~ \mathrm{ns}, v_p=3\times 10^5 ~\mathrm{m/s}, T=3\times 10^{11}~\mathrm{K}$, and $\beta = 33$
  • Figure 2: Transformation of a world line. For details see the text Appendix \ref{['app:cov']}.
  • Figure 3: Transformation of the coordinate frame and the complete world line. For details see the text Appendix \ref{['app:cov']}.
  • Figure 4: Division the domain of integration into parts. The overlap factor can be explained by the length of the red dashed line being the value of $\varphi_{n}$ at the same $v$.
  • Figure :
  • ...and 2 more figures