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Principle of maximum caliber

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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, 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

Contents

S [ ρ [ x ( ) ] ] = ∫ D x ρ [ x ( ) ] ln ⁡ ρ [ x ( ) ] π [ x ( ) ]

History

The principle of maximum caliber was proposed by Edwin T. Jaynes in 1980, in an article titled The Minimum Entropy Production Principle over the context of to find a principle for to derive the 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 is of the form

S [ ρ [ x ( ) ] ] = ∫ D x ρ [ x ( ) ] ln ⁡ ρ [ x ( ) ] π [ x ( ) ]

where for n-constraints

∫ D x ρ [ x ( ) ] A n [ x ( ) ] = ⟨ A n [ x ( ) ] ⟩ = a n

it is shown that the probability functional is

ρ [ x ( ) ] = exp ⁡ { − ∑ i = 0 n α n A n [ x ( ) ] } .

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

∫ D x ρ [ x ( ) ] L n ( x ( t ) , x ˙ ( t ) , t ) = ⟨ L n ( x ( t ) , x ˙ ( t ) , t ) ⟩ = ℓ ( 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 ) } .

Maximum caliber and statistical mechanics

Following the hypothesis of Jaynes, there are publications in which it appears as the principle of maximum caliber it is framed in the context of creating a statistical representation of systems with many degrees of freedom.

References

Principle of maximum caliber Wikipedia
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