Principle of maximum caliber

The principle of maximum caliber (MaxCal) or maximum path entropy principle, suggested by E. T. Jaynes,[1] 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

${\displaystyle S[\rho [x()]]=\int D_{x}\,\,\rho [x()]\,\ln {\rho [x()] \over \pi [x()]}}$

History

The principle of maximum caliber was proposed by Edwin T. Jaynes in 1980,[1] 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 ${\displaystyle S}$ is of the form

${\displaystyle S[\rho [x()]]=\int D_{x}\rho [x()]\ln {\rho [x()] \over \pi [x()]}}$

where for n-constraints

${\displaystyle \int D_{x}\rho [x()]A_{n}[x()]=\langle A_{n}[x()]\rangle =a_{n}}$

it is shown that the probability functional is

${\displaystyle \rho [x()]=\exp \left\{-\sum _{i=0}^{n}\alpha _{n}A_{n}[x()]\right\}.}$

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

${\displaystyle \int D_{x}\rho [x()]L_{n}(x(t),{\dot {x}}(t),t)=\langle L_{n}(x(t),{\dot {x}}(t),t)\rangle =\ell (t)}$

it is shown that the probability functional is

${\displaystyle \rho [x()]=\exp \left\{-\sum _{i=0}^{n}\int _{0}^{T}dt\,\alpha _{n}(t)L_{n}(x(t),{\dot {x}}(t),t)\right\}.}$

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.[2][3][4]

Notes

1. Jaynes, E. T. (1980) The Minimum Entropy Production Principle , Annu. Rev. Phys. Chem. 31, 579.
2. Steve Pressé, Kingshuk Ghosh, Julian Lee, and Ken A. Dill (2013), Principles of maximum entropy and maximum caliber in statistical physics , Rev. Mod. Phys. 85, 1115.
3. Michael J. Hazoglou, Valentin Walther, Purushottam D. Dixit and Ken A. Dill (2015), Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics, J. Chem. Phys. 143, 051104.
4. Davis S., González D. (2015), Hamiltonian formalism and path entropy maximization, Journal of Physics A-Mathematical and Theoretical Vol. 48 Num. 42