# V Anequilibrium Dynamics

[p. 2473l, §2]

The divergence of expectation values for block energies
makes it clear that the concept of stationarity
for macroscopic block variables requires modification
whenever the system is at an anequilibrium transition.
In the ensemble limit each block becomes an infinite system.
The present section discusses general aspects of the macroscopic dynamics.

[p. 2473l, §3]

The connection between the static properties discussed
in the preceding sections and the macroscopic time evolution
is provided by the ergodic hypothesis.
The ergodic hypothesis allows one to view
a sequence of block variables XiN as a sequence of snapshots XNti=XiN
representing a possible stroboscopically recorded time evolution of a single block.
This temporal embedding of an arbitrary sequence of block variables
defines the sequence ti of time points as a strictly increasing stochastic
sequence with stationary and positive random increments τi=ti-1-ti>0.
The time instants ti>0 are positive random variables and,
assuming t0=0, one has tN=τ1+…+τN.
The critical ensemble limit corresponds now
to the long-time limit of a large system.
In this limit the distribution function PNt of the variables
tN converges to that of a one-sided stable law with index ω~t,
and symmetry parameter ζt.
In general the index ω~t, will be a new exponent
which is different from the static exponents
and also depends on the macroscopic observable X of interest.

[p. 2473l, §4]

An important observation for the temporal process
defined through a sequence XiN of block variables is that
the limiting law must be one sided, i. e.,
the limiting distribution function Pt is nonzero only for t>0.
This places the restrictions ζt=1 and 0<ω~t≤1
on the parameters of the possible limiting distributions.
A second important

[p. 2473r, §4]

observation is that each distribution PNt
defines a stable convolution semigroup PDN [28]
with a continuous N-dependent parameter DN given in (3.18).
The convolution operators defined as

PDNft=∫-∞∞ft-t′dPNt′ | | (5.1) |

with DN=N/DL*N-11/ω~t, D>0 have the semigroup property [28]

PDNPDN′=PDN+DN′, | | (5.2) |

and this semigroup has a generator A defined by
A=limDN→0+DN-1PDN-I,
where I denotes the identity operator [28].
For sufficiently large N, PNt can be replaced
by a one-sided stable distribution.
The ergodic hypothesis allows to identify the generator of the semigroup (5.2)
with the generator for the time transformation of the macroscopic observable Xt.
The generators Aω~t of one-sided stable semigroups with index ω~t,
are well known [28] to be proportional
to the fractional time derivatives dω~t/dtω~t.
The case ω~t=1 is of special importance
because in this case the distribution PNt is degenerate,
the semigroup is a semigroup of translations and
the corresponding generator is the usual derivative d/dt.
The preceding observations imply that the definition of
stationarity for the time dependence of a macroscopic
block variable Xt must be generalized into

which reduces to the traditional definition d/dtXt=0
only for the special limiting case ω~t=1.
It is important to note that (5.3)
holds only for macroscopic block variables
but not for microscopic site variables.
The main result is that fractional linear differential operators appear
naturally as the generators of time transformations for
the observables of infinite systems by virtue of the ensemble limit
combined with the ergodic hypothesis.
The solution to (5.3) will in general be time dependent according to

where C0 is a constant.
Only for ω~t=1 does (5.4) reduce to constant Xt.
In general stationarity is already reached when Xt decays algebraically.
The main consequence of the generalized concept of stationarity expressed in (5.3)
is that algebraic time decays of macroscopic observables may in fact be stationary.

[p. 2473r, §5]

A further consequence of fractional time derivatives as
generators of the time evolution for macroscopic observables
is that the equations of motion for a macroscopic
observable Xt become generalized into

dω~tdtω~tXt=iLtXt, | | (5.5) |

where L denotes a generalized Liouville operator of the
system which may be explicitly time dependent.
For ω~t=1 and LtX=i/ℏHt,X reduces to the equations of motion
in the Heisenberg representation for a

[p. 2474l, §0]

system with a time-dependent Hamiltonian Ht.

[p. 2474l, §1]

Equation (5.5) has an interesting formal solution.
Laplace transformation yields the expression

Xu=uω~t-1uω~t-iL-1X0, | | (5.6) |

where X0=X(t=0), u denotes the Laplacian spectral variable,
and L has been assumed to be time independent.
Mellin transformation Mfs=∫0∞ftts-1dt gives the formal result

Xs=1ω~tΓs/ω~tΓ1-s/ω~tΓ1-siL-1s/ω~tX0, | | (5.7) |

where the relation Γ1-sMfs=MLfu1-s
between Mellin and Laplace transforms has been used.
Comparing the inverse Mellin transform with the
definition of the H-function given in the Appendix yields

X(t)=1ω~tH1,21,1((-iL)1/ω~tt|0,1ω~t0,1ω~t0,1)X0. | | (5.8) |

This result may be rewritten more conveniently by exploiting
the fact that the H-function H1,21,1 is closely related
to the class of Mittag-Leffler functions.
In this way the series representation

Xt=∑∞k=0tkω~tΓkωt+1iLkX0 | | (5.9) |

for the result (5.8) is obtained.
The result (5.9) describes the generalized solution
for nonstationary macroscopic observables whose macroscopic time evolution
is governed by L and for ω~=1
this is seen to reduce to the familiar result Xt=eiLtX0.
Evidently the solution (5.9)
represents very slow nonexponential dynamics
approaching algebraic time decay in the long-time limit.