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# 4 Fractional Stationarity

[554.1.1]This section investigates the condition of invariance or stationarity for the induced ultralong time dynamics . [554.1.2]Invariance of a measure on under the induced dynamics is defined as usual (see (3)) by requiring that

 (22)

for and . [554.1.3]For (22) may be called the condition of fractional invariance or fractional stationarity. [554.1.4]Using (2) the invariance condition becomes

 (23)

[page 555, §0]   for where is the infinitesimal generator of the semigroup . [555.0.1]For the relation (21) implies , and thus in this case invariant measures conserve volumes in phase space as usual. [555.0.2]A very different situation arises for .

[555.1.1]For the infinitesimal generators of the stable convolution semigroup are obtained [26] by evaluating the generalized function [29] on the time translation group

 (24)

where is a constant. [555.1.2]Comparing (24) with the Balakrishnan algorithm [30, 31, 32] for fractional powers of the generator of a semigroup

 (25)

shows that if denotes the infinitesimal generator of the original time evolution then is the infinitesimal generator of the induced time evolution . [555.1.3]For the generators for are fractional time derivatives [15, 31, 29]. [555.1.4]The differential form (23) of the fractional invariance condition for becomes

 (26)

for which was first derived in [18, 19]. [555.1.5]Its solution is

 (27)

for with a constant. [555.1.6]This shows that for a fractional stationary dynamical state is not constant. [555.1.7]Fractional stationarity or fractional invariance of a measure implies that phase space volumes shrink with time. [555.1.8]Thus fractional dynamics is dissipative. [555.1.9]More generally (26) reads with solution for in the sense of distributions. [555.1.10]The stationary solution with has a jump discontinuity at , and is not simply constant.

[555.2.1]The transition from an original invariant measure on to a fractional invariant measure on a subset of measure may be called stationarity breaking. [555.2.2]It occurs spontaneously in the sense that it is generated by the dynamics itself. [555.2.3]Stationarity breaking implies ergodicity breaking, and thus the ultralong time limit is a possible scenario for ergodicity breaking in ergodic theory.

[555.3.1]The present paper has shown that the use of fractional time derivatives in physics is not only justified, but arises generically for induced dynamics in the ultralong time limit. [555.3.2]This mathematical result applies to many physical situations. [555.3.3]In the simplest case the resulting fractional differential equation (26) defines fractional stationarity which provides the dynamical basis for the anequilibrium concept [18]. [555.3.4]Recently fractional random walks were discussed [8] and solved [10] in the continuum limit.

[page 556, §0]   ACKNOWLEDGEMENT : The author is grateful to Norges Forskningsrad (Nr.: 424.94 / 004 B) for financial support.