# Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking

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## Abstract

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## 1. Introduction

## 2. Derivation of the Jarzynski Equality

## 3. Models and Methods

#### 3.1. Single Oscillator under Linear Protocol

#### 3.2. Single Oscillator with Periodic Forcing

#### 3.3. Coupled Oscillators with Periodic Forcing

#### 3.4. Results

## 4. Irreversible Expansion of an Ideal Gas

## 5. Concluding Remarks

- A single oscillator pulled by a constant speed harmonic trap yields the infinite bath result if the process is not too fast. It sensibly and rapidly departs from that when the speed of the driving agent grows. The effect is more evident (as expected) for small than for large integration bounds, L and M, for smaller harmonic constants, and for smaller bath temperatures. In the infinite $L,M$ limits, the standard canonical result is recovered, but larger and larger L and M are required; the smaller are ${k}_{p}$ or $\beta $.
- For a single periodically driven oscillator, the infinite bath result over a multiple of the driving period equals 1. Strong deviations from this value, even those that reach 0, are found for a finite bath in finite intervals about the resonance frequency. Although the theoretical result is again obtained in the infinite $L,M$ limit, this is harder if the driving acts for longer times (i.e., for a larger number of periods).
- In the case in which the oscillator S is coupled to an oscillator E, the infinite bath value 1 is obtained, apart from oscillations, for a sufficiently large driving frequency. Noticeable deviations from that results are still present about specific values of the forcing frequency. In this case, we have no analytical expression for the finite bath result. Therefore this conclusion is based on numerical data for a finite ensemble of initial conditions, proven robust against variations of ensemble size.

Studying the thermodynamical state of a homogeneous fluid of a given volume at a given temperature […], we observe that there is an infinite number of states of molecular motion that correspond to it. With increasing time, the system exists successively in all the dynamical states that correspond to the given thermodynamical state. From this point of view, we may say that a thermodynamical state is the ensemble of all the dynamical states through which, as a result of the molecular motion, the system is rapidly passing.

If the transition mechanism among the atomic states is sufficiently effective, the system passes rapidly through all representative atomic states in the course of a macroscopic observation […]. However, under certain unique conditions, the mechanism of atomic transition may be ineffective, and the system may be trapped in a small subset of atypical atomic states. Or, even if the system is not completely trapped, the rate of transition may be so slow that a macroscopic measurement does not yield a proper average over all possible atomic states.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Some Explicit Calculations for Section 3.1

## Appendix B. Some Explicit Calculations for Section 3.2

## References

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**Figure 1.**Values of ${\u2329{e}^{-\beta {W}_{J}}\u232a}_{L,M}$ for the single harmonic oscillator with $\lambda =sin\gamma t$, as a function of the forcing frequency $\gamma $, for different values of ${\mathrm{\Gamma}}_{0}$ volumes and different final times $\tau $. Left and right panels refer to $L=M=1$ and $L=M=10$, respectively, with $\tau $ such that $B=sin\left(2\pi \right)$ for the first case, and $B=sin\left(200\pi \right)$ for the second. Blue, red, yellow, and purple plots refer to inverse temperatures $\beta =1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100,\phantom{\rule{0.277778em}{0ex}}1000$, respectively. Other parameters are set to $m=1$, $k=1$, ${k}_{D}=1$.

**Figure 2.**Values of ${\u2329{e}^{-\beta {W}_{J}}\u232a}_{L,M}$ for the single harmonic oscillator with $\lambda =sin\gamma t$, as a function of the forcing frequency $\gamma $, for different values of ${\mathrm{\Gamma}}_{0}$ volumes and different final times $\tau $. Left and right panels refer to $L=M=1$ and $L=M=10$, respectively, with $\tau $ such that $B=sin\left(2\pi \right)$ for the first case and $B=sin\left(200\pi \right)$ for the second. Blue, red, yellow, and purple plots refer to stiffnesses ${k}_{s}=0.1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100$, respectively. Other parameters are set to $m=1$, $\beta =1$, and ${k}_{D}=1$.

**Figure 3.**Values of ${\u2329{e}^{-\beta {W}_{J}}\u232a}_{L,M}$ for the single harmonic oscillator driven by a constant speed moving harmonic trap, with a final protocol value of $\lambda \left(\tau \right)=B=1$. The result is shown as a function of ℓ, for different values of L and M. Solid lines refer to $L=M=1$, dash-dotted lines to $L=M=2$, and dashed lines to $L=M=5$. In the left panel, blue, red, yellow, and purple plots refer to ${k}_{p}=0.1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100$, respectively. In the right panel, blue, red, yellow, and purple plots refer to $\beta =1,\phantom{\rule{0.277778em}{0ex}}4,\phantom{\rule{0.277778em}{0ex}}7,\phantom{\rule{0.277778em}{0ex}}10$, respectively. Other parameters are set to $m=1$ and ${k}_{D}=1$.

**Figure 4.**Behavior of ${\u2329{e}^{-\beta {W}_{J}}\u232a}_{L,M}$ as a function of the forcing frequency $\gamma $ for the coupled oscillators model, simulated with $\tau $ such that $B=sin\left(2\pi \right)$. Left, center, and right panels report the system behavior at varying values of the parameters $\beta $, ${k}_{S}$, and ${k}_{I}$, respectively. Markers represent results from numerical simulations, while dotted lines connecting them are a guide for the eye. Left panel: dark blue, light blue, light grey, and dark grey lines correspond to $\beta =0.1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100$, respectively; other parameters are set to ${m}_{S}={m}_{E}=1$, ${k}_{S}={k}_{E}=1$, ${k}_{I}=1$ and $L=M=1$. Center panel: dark blue, light blue, light grey, and dark grey lines correspond to ${k}_{S}=0.1,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100$, respectively; other parameters are set to ${m}_{S}={m}_{E}=1$, ${k}_{E}=1$, ${k}_{I}=1$, $\beta =1$ and $L=M=1$. Right panel, dark blue, and light blue lines refer to ${k}_{I}=10$ and ${k}_{I}=0.1$, respectively, with $L=M=1$; dark grey and light grey plots refer to ${k}_{I}=10$ and ${k}_{I}=0.1$, respectively; for $L=M=5$, the remaining parameters are set to ${m}_{E}=10$, ${m}_{S}=1$, ${k}_{S}={k}_{E}=1$, $\beta =1$.

**Figure 5.**Schematic representation of the irreversible expansion of an ideal gas. A 2D box contains non-interacting particles in equilibrium with a heat bath at temperature T. Panel (

**a**): state of the system before the central wall is removed; all the particles are confined in the left half of the box and undergo specular reflections with the container walls. Panel (

**b**): dynamics of the particles once the central wall is removed. Panel (

**c**): after the wall is reintroduced, a fraction of particles is trapped in the right half of the box.

**Figure 6.**Left panel: behavior of ${N}_{R}/N$ as a function of the protocol duration $\tau $, for different values of $\beta $, with $N=10,000$, $L=5$, $m=1$. Blue, orange, and yellow solid lines refer to Equation (39) for $\beta =1,\phantom{\rule{0.277778em}{0ex}}10,\phantom{\rule{0.277778em}{0ex}}100$, respectively, and are obtained by truncating the Formula (39) to $n=1000$. The black dotted lines denote the results of the numerical simulations. Dash-dotted vertical lines indicate the characteristic times obtained from Equation (40). Right panel: behavior of ${N}_{R}/N$ vs. $\tau $, for different values of N, obtained from numerical simulations. Black, blue, and orange solid lines refer to $N=20,\phantom{\rule{0.277778em}{0ex}}100,\phantom{\rule{0.277778em}{0ex}}500$, respectively. The other parameters are fixed to $L=5$, $\beta =10$, $m=1$.

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**MDPI and ACS Style**

Colangeli, M.; Di Francesco, A.; Rondoni, L.
Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking. *Symmetry* **2023**, *15*, 1268.
https://doi.org/10.3390/sym15061268

**AMA Style**

Colangeli M, Di Francesco A, Rondoni L.
Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking. *Symmetry*. 2023; 15(6):1268.
https://doi.org/10.3390/sym15061268

**Chicago/Turabian Style**

Colangeli, Matteo, Antonio Di Francesco, and Lamberto Rondoni.
2023. "Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking" *Symmetry* 15, no. 6: 1268.
https://doi.org/10.3390/sym15061268