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Principle of maximum caliber

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Statistical principle

The principle of maximum caliber (MaxCal) or maximum path entropy principle, suggested by E. T. Jaynes, can be considered as a generalization of the principle of maximum entropy. It postulates that the most unbiased probability distribution of paths is the one that maximizes their Shannon entropy. This entropy of paths is sometimes called the "caliber" of the system, and is given by the path integral

S [ ρ [ x ( ) ] ] = ∫ D x ρ [ x ( ) ] ln ⁡ ρ [ x ( ) ] π [ x ( ) ] {\displaystyle S]=\int D_{x}\,\,\rho \,\ln {\rho \over \pi }}

History

The principle of maximum caliber was proposed by Edwin T. Jaynes in 1980, in an article titled The Minimum Entropy Production Principle in the context of deriving a principle for non-equilibrium statistical mechanics.

Mathematical formulation

The principle of maximum caliber can be considered as a generalization of the principle of maximum entropy defined over the paths space, the caliber S {\displaystyle S} is of the form

S [ ρ [ x ( ) ] ] = ∫ D x ρ [ x ( ) ] ln ⁡ ρ [ x ( ) ] π [ x ( ) ] {\displaystyle S]=\int D_{x}\rho \ln {\rho \over \pi }}

where for n-constraints

∫ D x ρ [ x ( ) ] A n [ x ( ) ] = ⟨ A n [ x ( ) ] ⟩ = a n {\displaystyle \int D_{x}\rho A_{n}=\langle A_{n}\rangle =a_{n}}

it is shown that the probability functional is

ρ [ x ( ) ] = exp ⁡ { − ∑ i = 0 n α n A n [ x ( ) ] } . {\displaystyle \rho =\exp \left\{-\sum _{i=0}^{n}\alpha _{n}A_{n}\right\}.}

In the same way, for n dynamical constraints defined in the interval t ∈ [ 0 , T ] {\displaystyle t\in } of the form

∫ D x ρ [ x ( ) ] L n ( x ( t ) , x ˙ ( t ) , t ) = ⟨ L n ( x ( t ) , x ˙ ( t ) , t ) ⟩ = ℓ ( t ) {\displaystyle \int D_{x}\rho L_{n}(x(t),{\dot {x}}(t),t)=\langle L_{n}(x(t),{\dot {x}}(t),t)\rangle =\ell (t)}

it is shown that the probability functional is

ρ [ x ( ) ] = exp ⁡ { − ∑ i = 0 n ∫ 0 T d t α n ( t ) L n ( x ( t ) , x ˙ ( t ) , t ) } . {\displaystyle \rho =\exp \left\{-\sum _{i=0}^{n}\int _{0}^{T}dt\,\alpha _{n}(t)L_{n}(x(t),{\dot {x}}(t),t)\right\}.}

Maximum caliber and statistical mechanics

Following Jaynes' hypothesis, there exist publications in which the principle of maximum caliber appears to emerge as a result of the construction of a framework which describes a statistical representation of systems with many degrees of freedom.

See also

Notes

  1. ^ Jaynes, E T (1980). "The Minimum Entropy Production Principle". Annual Review of Physical Chemistry. 31 (1). Annual Reviews: 579–601. doi:10.1146/annurev.pc.31.100180.003051. ISSN 0066-426X.
  2. Pressé, Steve; Ghosh, Kingshuk; Lee, Julian; Dill, Ken A. (2013-07-16). "Principles of maximum entropy and maximum caliber in statistical physics". Reviews of Modern Physics. 85 (3). American Physical Society (APS): 1115–1141. doi:10.1103/revmodphys.85.1115. ISSN 0034-6861.
  3. Hazoglou, Michael J.; Walther, Valentin; Dixit, Purushottam D.; Dill, Ken A. (2015-08-06). "Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics". The Journal of Chemical Physics. 143 (5). AIP Publishing: 051104. arXiv:1505.05479. doi:10.1063/1.4928193. ISSN 0021-9606.
  4. Davis, Sergio; González, Diego (2015-09-22). "Hamiltonian formalism and path entropy maximization". Journal of Physics A: Mathematical and Theoretical. 48 (42). IOP Publishing: 425003. arXiv:1404.3249. doi:10.1088/1751-8113/48/42/425003. ISSN 1751-8113.
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