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 $X_{{iN}}$ as a sequence of snapshots $X_{N}(t_{i})=X_{{iN}}$ representing a possible stroboscopically recorded time evolution of a single block. This temporal embedding of an arbitrary sequence of block variables defines the sequence $t_{i}$ of time points as a strictly increasing stochastic sequence with stationary and positive random increments $\tau _{i}=t_{{i-1}}-t_{i}>0$ . The time instants $t_{i}>0$ are positive random variables and, assuming $t_{0}=0$ , one has $t_{N}=\tau _{1}+\dots+\tau _{N}$ . The critical ensemble limit corresponds now to the long-time limit of a large system. In this limit the distribution function $P_{N}(t)$ of the variables $t_{N}$ converges to that of a one-sided stable law with index ${\tilde{\omega}}_{t}$ , and symmetry parameter $\zeta _{t}$ . In general the index ${\tilde{\omega}}_{t}$ , 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 $X_{{iN}}$ of block variables is that the limiting law must be one sided, i. e., the limiting distribution function $P(t)$ is nonzero only for $t>0$ . This places the restrictions $\zeta _{t}=1$ and $0<{\tilde{\omega}}_{t}\leq 1$ on the parameters of the possible limiting distributions. A second important

[p. 2473r, §4]
observation is that each distribution $P_{N}(t)$ defines a stable convolution semigroup $\mathfrak{P}(D_{N})$ [28] with a continuous $N$ -dependent parameter $D_{N}$ given in (3.18). The convolution operators defined as

$\mathfrak{P}(D_{N})f(t)=\int^{\infty}_{{-\infty}}f(t-t^{\prime})dP_{N}(t^{\prime})$

(5.1)

with $D_{N}=[N/D\mathfrak{L}^{*}(N^{{-1}})]^{{1/{\tilde{\omega}}_{t}}}$ , $D>0$ have the semigroup property [28]

$\mathfrak{P}(D_{N})\mathfrak{P}(D^{\prime}_{N})=\mathfrak{P}(D_{N}+D^{\prime}_{N}),$

(5.2)

and this semigroup has a generator $\mathfrak{A}$ defined by $\mathfrak{A}=\lim _{{D_{N}\rightarrow 0^{+}}}D^{{-1}}_{N}[\mathfrak{P}(D_{N})-\mathfrak{I}]$ , where $\mathfrak{I}$ denotes the identity operator [28]. For sufficiently large $N$ , $P_{N}(t)$ 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 $X(t)$ . The generators $\mathfrak{A}_{{{\tilde{\omega}}_{t}}}$ of one-sided stable semigroups with index ${\tilde{\omega}}_{t}$ , are well known [28] to be proportional to the fractional time derivatives $d^{{{\tilde{\omega}}_{t}}}/dt^{{{\tilde{\omega}}_{t}}}$ . The case ${\tilde{\omega}}_{t}=1$ is of special importance because in this case the distribution $P_{N}(t)$ 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 $X(t)$ must be generalized into

$\frac{d^{{{\tilde{\omega}}_{t}}}}{dt^{{{\tilde{\omega}}_{t}}}}X(t)=0$

(5.3)

which reduces to the traditional definition $(d/dt)X(t)=0$ only for the special limiting case ${\tilde{\omega}}_{t}=1$ . 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

$X(t)=C_{0}t^{{{\tilde{\omega}}_{t}-1}},$

(5.4)

where $C_{0}$ is a constant. Only for ${\tilde{\omega}}_{t}=1$ does (5.4) reduce to constant $X(t)$ . In general stationarity is already reached when $X(t)$ 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 $X(t)$ become generalized into

$\frac{d^{{{\tilde{\omega}}_{t}}}}{dt^{{{\tilde{\omega}}_{t}}}}X(t)=i\mathscr L(t)X(t),$

(5.5)

where $\mathscr L$ denotes a generalized Liouville operator of the system which may be explicitly time dependent. For ${\tilde{\omega}}_{t}=1$ and $\mathscr L(t)X=(i/\hbar)[\mathscr H(t),X]$ reduces to the equations of motion in the Heisenberg representation for a

[p. 2474l, §0]
system with a time-dependent Hamiltonian $\mathscr H(t)$ .

[p. 2474l, §1]
Equation (5.5) has an interesting formal solution. Laplace transformation yields the expression

$X(u)=u^{{{\tilde{\omega}}_{t}-1}}(u^{{{\tilde{\omega}}_{t}}}-i\mathscr L)^{{-1}}X_{0},$

(5.6)

where $X_{0}=X(t=0)$ , $u$ denotes the Laplacian spectral variable, and $\mathscr L$ has been assumed to be time independent. Mellin transformation $\mathfrak{M}\{ f\}(s)=\int^{\infty}_{0}f(t)t^{{s-1}}dt$ gives the formal result

$X(s)=\frac{1}{{\tilde{\omega}}_{t}}\frac{\Gamma(s/{\tilde{\omega}}_{t})\Gamma(1-(s/{\tilde{\omega}}_{t}))}{\Gamma(1-s)}(i\mathscr L^{{-1}})^{{s/{{\tilde{\omega}}_{t}}}}X_{0},$

(5.7)

where the relation $\Gamma(1-s)\mathfrak{M}\{ f\}(s)=\mathfrak{M}\{\mathfrak{L}\{ f\}(u)\}(1-s)$ 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

$X(t)=\frac{1}{{\tilde{\omega}}_{t}}H^{{1,1}}_{{1,2}}\left((-i\mathscr L)^{{1/{{\tilde{\omega}}_{t}}}}t\,\middle|\begin{array}[]{l}\big(0,\frac{1}{{\tilde{\omega}}_{t}}\big)\\ \big(0,\frac{1}{{\tilde{\omega}}_{t}}\big)(0,1)\end{array}\right)X_{0}.$

(5.8)

This result may be rewritten more conveniently by exploiting the fact that the $H$ -function $H^{{1,1}}_{{1,2}}$ is closely related to the class of Mittag-Leffler functions. In this way the series representation

$X(t)=\left(\sum^{\infty}_{{k=0}}\frac{t^{{k{\tilde{\omega}}_{t}}}}{\Gamma(k\omega _{t}+1)}(i\mathscr L)^{k}\right)X_{0}$

(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 $\mathscr L$ and for ${\tilde{\omega}}=1$ this is seen to reduce to the familiar result $X(t)=e^{{i\mathscr Lt}}X_{0}$ . Evidently the solution (5.9) represents very slow nonexponential dynamics approaching algebraic time decay in the long-time limit.

Authors:	R. Hilfer
originally published in:	Physical Review E48, No. 2, pages 2466-2475 (1993)
originally submitted:	January 29th, 1993