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Fluctuation theorem
(section)
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== Summary == The fluctuation theorem is of fundamental importance to [[non-equilibrium statistical mechanics]]. The FT (together with the [[universal causation]] proposition) gives a generalisation of the [[second law of thermodynamics]] which includes as a special case, the conventional second law. It is then easy to prove the Second Law Inequality and the NonEquilibrium Partition Identity. When combined with the [[central limit theorem]], the FT also implies the [[Green-Kubo relations]] for linear transport coefficients, close to equilibrium. The FT is however, more general than the Green-Kubo Relations because unlike them, the FT applies to fluctuations far from equilibrium. In spite of this fact, scientists have not yet been able to derive the equations for nonlinear response theory from the FT. The FT does '''not''' imply or require that the distribution of time averaged dissipation be Gaussian. There are many examples known where the distribution of time averaged dissipation is non-Gaussian and yet the FT (of course) still correctly describes the probability ratios. Lastly the theoretical constructs used to prove the FT can be applied to ''nonequilibrium transitions'' between two different ''equilibrium'' states. When this is done the so-called [[Jarzynski equality]] or nonequilibrium work relation, can be derived. This equality shows how equilibrium free energy differences can be computed or measured (in the laboratory<ref>{{cite journal |last1=Rademacher |first1=Markus |last2=Konopik |first2=Michael |last3=Debiossac |first3=Maxime |last4=Grass |first4=David |last5=Lutz |first5=Eric |last6=Kiesel |first6=Nikolai |title=Nonequilibrium Control of Thermal and Mechanical Changes in a Levitated System |journal=Physical Review Letters |date=15 February 2022 |volume=128 |issue=7 |pages=070601 |doi=10.1103/physrevlett.128.070601|pmid=35244419 |arxiv=2103.10898 |bibcode=2022PhRvL.128g0601R |s2cid=232290453 }}</ref>), from nonequilibrium path integrals. Previously quasi-static (equilibrium) paths were required. The reason why the fluctuation theorem is so fundamental is that its proof requires so little. It requires: * knowledge of the mathematical form of the initial distribution of molecular states, * that all time evolved final states at time ''t'', must be present with nonzero probability in the distribution of initial states (''t'' = 0) β the so-called condition of ''ergodic consistency'' and * an assumption of time reversal symmetry. In regard to the latter "assumption", while the equations of motion of quantum dynamics may be time-reversible, quantum processes are nondeterministic by nature. What state a wave function collapses into cannot be predicted mathematically, and further the unpredictability of a quantum system comes not from the myopia of an observer's perception, but on the intrinsically nondeterministic nature of the system itself. In [[physics]], the [[Newton's laws of motion|laws of motion]] of [[classical mechanics]] exhibit time reversibility, as long as the operator Ο reverses the [[conjugate momentum|conjugate momenta]] of all the particles of the system, i.e. <math>\mathbf{p} \rightarrow \mathbf{-p} </math> ([[T-symmetry]]). In [[quantum mechanics|quantum mechanical]] systems, however, the [[weak nuclear force]] is not invariant under T-symmetry alone; if weak interactions are present reversible dynamics are still possible, but only if the operator Ο also reverses the signs of all the [[charge (physics)|charges]] and the [[parity (physics)|parity]] of the spatial co-ordinates ([[C-symmetry]] and [[P-symmetry]]). This reversibility of several linked properties is known as [[CPT symmetry]]. [[Thermodynamic process]]es can be [[Reversible process (thermodynamics)|reversible]] or [[irreversible process|irreversible]], depending on the change in [[entropy]] during the process.
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