RECORD
Duration of local violations of the second law of thermodynamics along single trajectories in phase space
Statistical Mechanics · Submitted 4 Dec 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We define the {\it violation fraction} $\nu$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $\tau$. In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. The exponent $\alpha$ is positive and generally depends on the protocol for the external driving forces, being $\alpha=1$ for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of $\langle \nu \rangle$ with $N$ and $\tau$ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We define the {\it violation fraction} $\nu$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. | contribution 0.69 | 0.85 | 0.04 | asserted 0.98 |
| s1 | This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $\tau$. | background 0.44 | 0.50 | 0.05 | asserted 1.00 |
| s2 | In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. | result 1.00 | 0.94 | 0.81 | asserted 1.00 |
| s3 | The exponent $\alpha$ is positive and generally depends on the protocol for the external driving forces, being $\alpha=1$ for a constant drive. | result 0.92 | 0.69 | 0.13 | asserted 0.98 |
| s4 | As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. | method 0.52 | 0.90 | 0.03 | asserted 0.95 |
| s5 | In this case we show that the scaling of $\langle \nu \rangle$ with $N$ and $\tau$ agrees with our result. | result 0.94 | 0.95 | 0.09 | asserted 1.00 |
| s6 | Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition. | limitation 0.72 | 0.48 | 0.05 | hedged possibility 0.98 |
What was examined, and what was not
- arXiv listing abstract v5
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We define the {\it violation fraction} $\nu$ as the cumulative fraction of time that the entropy change is negative during single realizations of processes in phase space. This quantity depends both on the number of degrees of freedom $N$ and the duration of the time interval $\tau$. In the large-$\tau$ and large-$N$ limit we show that, for ergodic and microreversible systems, the mean value of $\nu$ scales as $\langle\nu(N,\tau)\rangle\sim\big(\tau N^{\frac{1}{1+\alpha}}\big)^{-1}$. The exponent $\alpha$ is positive and generally depends on the protocol for the external driving forces, being $\alpha=1$ for a constant drive. As an example, we study a nontrivial model where the fluctuations of the entropy production are non-Gaussian: an elastic line driven at a constant rate by an anharmonic trap. In this case we show that the scaling of $\langle \nu \rangle$ with $N$ and $\tau$ agrees with our result. Finally, we discuss how this scaling law may break down in the vicinity of a continuous phase transition.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.