Fluctuation theorem
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The second law of thermodynamics stands in apparent contradiction with the time reversible equations of motion for classical and quantum systems. This is often referred to as*Loschmidt's paradox. The fluctuation theorem (FT) gives a resolution to this "paradox".
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Statement of the fluctuation theorem (roughly)
This theorem (FT) gives a mathematical expression for the probability ratio that time-averaged irreversible entropy production[1], <math>\overline{\Sigma}_t<math> takes on a value, A, to the opposite value, −A, in systems away from equilibrium. In other words, for a finite non-equilibrium system in a finite time, the FT gives the probability that entropy will flow in a direction opposite to that dictated by the second law of thermodynamics.
Mathematically, the FT is expressed as:
- <math> \frac{\Pr(\overline{\Sigma}_{t}=A)}{\Pr(\overline{\Sigma}_{t}=-A)}=e^{At} <math>
This means that as the time or system size increases (since <math>\Sigma<math> is extensive), the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. The FT is one of the few expression in non-equilibrium statistical mechanics that is valid far from equilibrium.
The FT was first introduced by Evans (http://rsc.anu.edu.au/~evans/), Cohen and Morriss in 1993 in the journal Physical Review Letters. Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles. Recently, the first laboratory experiment was performed that verified the validity of the FT. In this experiment, a plastic bead was pulled through a solution by a laser. Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems. See Wang et al. [Phys Rev Lett, 89, 050601(2002)] and later Carberry et al, [Phys Rev Lett, 92, 140601(2004)]. This work was widely reported in the press - Second law of thermodynamics "broken" (NewScientist, 19 July 2002) (http://www.newscientist.com/article.ns?id=dn2572).
Note that the FT does not state that the second law of thermodynamics is wrong or invalid. The second law of thermodynamics is a statement about macroscopic systems. The FT is more general. It can be applied to both microscopic and macroscopic systems. When applied to macroscopic systems, the FT verifies the second law of thermodynamics.
Second law inequality
A simple consequence of the fluctuation theorem given above is that if we carry out an ensemble of experiments from some initial time t=0, and perform an ensemble average of time averages of the entropy production then an exact consequence of the FT is that the ensemble average cannot be negative for any value of the averaging time t:
- <math> \left\langle {\overline \Sigma _t } \right\rangle \ge 0,\quad \forall t <math>
For rather obvious reasons this equality is called the second law inequality [Searles & Evans, Aust J Chem, 57, 1119 (2004)].
Kawasaki identity
Another remarkably simple and elegant consequence of the FT is the so-called Kawasaki identity:
- <math> \left\langle {\exp [ - \overline \Sigma_t \; t ]} \right\rangle = 1,\quad \forall t <math>
see Carberry et al J Chem Phys 121, 8179(2004). Thus in spite of the Second Law Inequality which might lead you to expect that the Kawasaki average would decay exponentially with time, the exponential probability ratio given by the FT exactly cancels the negative exponential in the Kawasaki average above leading to a Kawasaki average which is unity for all time!
There are many important implications from the FT. One is that small machines (such as nanomachines or even mitochondria in a cell) will spend part of their time actually running in "reverse". By "reverse", it is meant that they function so as to run in a way opposite to that for which they were presumably designed. As an example, consider a jet engine. If a jet engine were to run in "reverse" in this context, it would take in ambient heat and exhaust fumes to generate kerosene and oxygen.
Dissipation function
[1] Strictly speaking the fluctuation theorem refers to a quantity known as the dissipation function. In thermostated nonequilibrium states the long time average of the dissipation function is equal to the average entropy production. However the FT refers to fluctuations rather than averages. The dissipation function is defined as,
- <math>
\Omega _t (\Gamma ) = \int_0^t {ds\;\Omega (\Gamma ;s)} \equiv \ln \left[ {\frac{{f(\Gamma ,0)}}{{f(\Gamma (t),0)}}} \right] - \frac{{\Delta Q(\Gamma ;t)}}{kT}
<math>
where k is Boltzmann's constant, <math>f(\Gamma , 0)<math> is the initial (t = 0) distribution of molecular states <math>\Gamma <math>, and <math> \Gamma (t) <math> is the molecular state arrived at after time t, under the exact time reversible equations of motion. <math> f(\Gamma (t),0) <math> is the INITIAL distribution of those time evolved states.
Note: in order for the FT to be valid we require that <math>f(\Gamma (t),0) \ne 0,\;\forall \Gamma (0) <math>. This condition is known as the condition of ergodic consistency. It is widely satisfied in common statistical ensembles - e.g. the canonical ensemble.
The system may be in contact with a large heat reservoir in order to thermostat the system of interest. If this is the case <math> \Delta Q(t) <math> is the heat lost to the reservoir over the time (0,t) and T is the absolute equilibrium temperature of the reservoir - see Williams et al, Phys Rev E70, 066113(2004). With this definition of the dissipation function the precise statement of the FT simply replaces entropy production with the dissipation function in each of the FT equations above.
Example: If one considers electrical conduction across a cell in contact with a large heat reservoir at temperature T, then the dissipation function is
- <math>
\Omega = - JF_e V/{kT}
<math>
the total electric current density J multiplied by the voltage drop across the circuit, <math>F_e <math>, and the system volume V , divided by the absolute temperature T, of the heat reservoir times Boltzmann's constant. Thus the dissipation function is easily recognised as the Ohmic work done on the system divided by the temperature of the reservoir. The long time average of this quantity is equal to the average spontaneous entropy production per unit time - see de Groot and Mazur "Nonequilibrium Thermodynamics" (Dover), equation (61), page 348.
Summary
The fluctuation theorem is of fundamental importance to nonequilibrium statistical mechanics. The FT (together with the axiom of causality) 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 Kawasaki identity. When combined with the central limit theorem, the FT also implies the famous 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 is 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 been 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), 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:
- a 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) - the so-called condition of ergodic consistency and,
- it requires time reversal symmetry.
Related topics
- Loschmidt's paradox
- Jarzynski equality - another nonequilibrium equality closely related to the fluctuation theorem and to the second law of thermodynamics
- Green-Kubo relations - there is a deep connection between the fluctuation theorem and the Green-Kubo relations for linear transport coefficients - like shear viscosity or thermal conductivity
- Boltzmann
- Thermodynamics