Principle of maximum caliber

The principle of maximum caliber was proposed by Edwin T. Jaynes in 1980,[1] in an article titled The Minimum Entropy Production Principle in the context of deriving a principle for non-equilibrium statistical mechanics.

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

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

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]