Abstract
The fluctuation theorem contains almost all general relations in thermodynamics and nonequilibrium statistical mechanics, including the second law of thermodynamics and the fluctuation-dissipation theorem, as its corollary.
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Notes
- 1.
We here use Jensen’s inequality for continuous variables.
- 2.
Since we have not assumed phenomenological results in thermodynamics here, we cannot presuppose the phenomenological fact that macroscopic observables in a thermodynamic process do not have macroscopic fluctuation.
- 3.
This theorem is a very rough evaluation. A stronger and refined result with a similar motivation is seen in Ref. [64].
- 4.
In the conventional fluctuation-dissipation theorem, this assumption is unnecessary. In Sect. 6.2.4, we discuss the difference between the conventional FDT and that derived from the fluctuation theorem.
- 5.
Precisely speaking, \(\hat{J}_X(t)\) is defined as an instant current of X between a system and one of the baths at time t. Although this definition depends on the choice of baths, the value in the long time limit is irrelevant to the measurement position.
- 6.
Another example of the conjugate intensive variable in terms of energy is \(P_N=\mu \) (chemical potential), and another example of the conjugate intensive variable in terms of entropy is \(\Pi _N=-\beta \mu =-\mu /T\). In general, if X is not energy, \(P_X=-T\Pi _X\) is satisfied.
- 7.
The product of the conductivity and the current is the amount of dissipation (entropy production). This is why Eq. (6.9) is called the fluctuation-dissipation theorem. When an external field does not induce dissipation and the final stationary state after the addition of the external field is still in equilibrium, some people call the same relation as the fluctuation response relation
- 8.
Precisely, the term \( \Delta \hat{s}\) disappears in Eq. (6.15) through the multiplication \(1/\tau \) and taking the long-time limit \(\tau \rightarrow \infty \). We note that \(\langle \Delta \hat{s}\rangle =0\) in stationary states, and thus we assert \(\lim _{\tau \rightarrow \infty }\langle \int _0^\tau dt \hat{J}(t) \Delta \hat{s}\rangle /\tau =0\).
- 9.
The dependence on \( \Delta \Pi _X\) comes through \( \Delta \Pi _X\)-dependence of \(P(\Gamma )\).
- 10.
This is easily realized in the case of voltage oscillation. To realize such oscillation in general setups, we prepare infinitely many baths with slightly different \(\Pi _X\)’s and attach and detach these baths so that the change in \(\Pi _X\) mimics \(\Pi _X (t)\).
- 11.
We used \( \Delta s =O(1)\) and \( \Delta s^2=O(1)\) in terms of \(\tau \).
- 12.
We again assumed that \(\left\langle \Delta \hat{J}(t) \Delta \hat{J}(0)\right\rangle _0\) converges rapidly to zero as t becomes large.
- 13.
The original proof of Saito and Utsumi [66] adopts a different approach from that we have seen here, the method of generating functions.
- 14.
For the case with fixed driving (zero frequency), the stationary condition confirms that we observe the same amount of stationary current at any position in the system. In contrast, for the case with oscillating driving (finite frequency), the amount of oscillating current may depend on the measurement position in the system (e.g., the current at the edge of the system and that in bulk may differ).
- 15.
Our argument here is restricted to classical systems. In the quantum case, the Kubo formula is expressed with the canonical conjugate, which is not directly obtained through the FT.
- 16.
In the conventional explanation of the Kubo formula (e.g., Ref. [32]), this difference is discussed as follows: We first notice that thermodynamic and mechanical forces should balance out in an equilibrium state. On the basis of this fact, we assert that these two forces can also be connected in steady state in a close-to-equilibrium regime.
- 17.
We remark that this setup is still, however, different from that in the Kubo formula. In the Kubo formula, we assume that the system is not attached to a heat bath except in the infinite past. In contrast, we suppose that the system is attached to a heat bath, which is a more realistic setup.
- 18.
These four baths are not necessarily separated physically. In the usual setup of thermoelectric transport, one of the heat baths and one of the particle baths are realized by a single heat-particle bath.
- 19.
In a canonical distribution, there exists a state accompanying a finite current with an extremely small probability.
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Shiraishi, N. (2023). Reduction from Fluctuation Theorem to Other Thermodynamic Relations. In: An Introduction to Stochastic Thermodynamics. Fundamental Theories of Physics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-19-8186-9_6
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DOI: https://doi.org/10.1007/978-981-19-8186-9_6
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