Response Function Theory for Many-Body Systems Away from Equilibrium: Conditions of Ultrafast-Time and Ultrasmall-Space Experimental Resolution

Abstract

A response function theory and scattering theory applicable to the study of physical properties of systems driven arbitrarily far removed from equilibrium, specialized for dealing with ultrafast processes, and in conditions of space resolution (including the nanometric scale) are presented. The derivation is done in the framework of a Gibbs-style nonequilibrium statistical ensemble formalism. The observable properties are shown to be connected with time- and space-dependent correlation functions out of equilibrium. A generalized fluctuation-dissipation theorem, which relates these correlation functions with generalized susceptibilities, is derived. The method of nonequilibrium-thermodynamic Green functions, which proves useful for calculations, is also presented. Two illustrative applications of the formalism, which study optical responses in ultrafast laser spectroscopy and Raman scattering of electrons in III-N semiconductors (of “blue diodes”) driven away from equilibrium by electric fields of moderate to high intensities, are described.

Appendix: The Nonequilibrium Statistical Operator

The construction of nonequilibrium statistical ensembles, that is, a nonequilibrium statistical ensemble formalism, NESEF for short [8–13], basically consisting of deriving a nonequilibrium statistical operator (probability distribution in the classical case) has been attempted along several lines. In a brief summarized way, we describe the construction of NESEF within a heuristic approach. In this context, several important points must be carefully taken into account in each case under consideration:

• 1. Choice of the basic variables. The choices for equilibrium and for nonequilibrium are different; for systems in equilibrium, it suffices to consider the constants of motion. To choose the variables for systems out of equilibrium, we must identify macroscopic processes and measurements and focus attention not only on the observables, but also on the character and expectations concerning the evolution equations for these variables [12, 13, 90]. Although all observables—along with their correlations, in certain cases—must be dealt with at the initial stage, as time elapses more and more contracted descriptions become possible as Bogoliubov’s principle of correlation weakening and the accompanying hierarchy of relaxation times comes into play [91].

• 2. Irreversibility (or Eddington’s arrow of time). Concerning this facet of the problem, Rudolf Peierls has stated that “In any theoretical treatment of transport problems, it is important to realize at what point the irreversibility has been incorporated. If it has not been incorporated, the treatment is wrong. A description of the situation that preserves the reversibility in time is bound to give the answer zero or infinity for any conductivity. If we do not see clearly where the irreversibility is introduced, we do not clearly understand what we are doing” [92].

• 3. Historicity. Proper treatment must incorporate the entire past dynamics of the system, or historicity effects, from a starting description of the macrostate of the sample in the experiment under study, say at t ₀, up to the time t when a measurement is performed. This is a quite important point in the case of dissipative systems, as emphasized among others by John Kirkwood, Melvin Green, Robert Zwanzig, and Hazime More [36–40]. It implies that the history of the system is not merely the series of events involving the system, but the series of transformations along time through which the system progressively comes into being at the measurement time t, through the evolution governed by the laws of mechanics [93]

Concerning the choice of the basic variables, in contrast with the equilibrium case, immediately after the open N-particle system, in contact with external sources and reservoirs, has been driven out of equilibrium, it becomes necessary to describe its state in terms of all its observables and, in certain instances, to introduce direct and cross-correlations. As time elapses, however, Bogoliubov’s principle of correlation weakening allows us to introduce increasing contractions of descriptions.

On the question of irreversibility, Nicolai S. Krylov [42] considered that there always exists a physical interaction between the measured system and the external world that is constantly “jolting” the system out of its exact microstate. Thus, the instability of trajectories and the unavoidable finite interaction with the outside would guarantee the validity of a “crudely prepared” macroscopic description. In the absence of a proper way to introduce this effect, one needs to resort to the interventionist’s approach, which is grounded on this ineluctable randomization process and leads to the asymmetric evolution of the macrostate.

The “intervention” consists of introducing in the Liouville equation for the statistical operator of the otherwise isolated system, a special source accounting for Krylov’s “jolting” effect, in the following form (for the logarithm of the statistical operator):

$$ \frac{\partial}{\partial t}\ln \mathcal{R}_{\varepsilon}(t) + \frac{1}{i \hbar}[\ln \mathcal{R}_{\varepsilon}(t),\hat{H}] = - \varepsilon \lbrack \ln \mathcal{R}_{\varepsilon}(t) - \ln \bar{\mathcal{R}}(t,0)], $$

(211)

where ε (effectively the reciprocal of a relaxation time) is taken to go to +0 after the average values have been computed.

This mathematically inhomogeneous term, in the otherwise normal Liouville equation, implies a continuous tendency to relax the statistical operators toward a referential distribution, $\bar {\mathcal {R}}$, which, as discussed below, represents an instantaneous quasi-equilibrium condition.

We can regard (211) as a regular Liouville equation with an infinitesimal source, which provides Bogoliubov’s symmetry breaking of time reversal and justifies disregarding the advanced solutions [8, 12, 13, 94, 95]. This is described by a Poisson distribution and the result at time t is obtained by averaging over all $t^{\prime }$ in the interval (t ₀,t), since the solution of (211) is

$$ \mathcal{R}_{\varepsilon}(t) = \exp \left\{ -\hat{S}(t,0) + \int \limits_{t_{0}}^{t}dt^{\prime }e^{\varepsilon (t^{\prime }-t)}\frac{d}{dt^{\prime }}\hat{S}(t^{\prime },t^{\prime }-t)\right\} , $$

(212)

where

$$ \hat{S}(t,0) = -\ln \bar{\mathcal{R}}(t,0), $$

(213)

$$\begin{array}{@{}rcl@{}} \hat{S}(t^{\prime },t^{\prime }-t) &=& \exp \left\{ -\frac{1}{i\hbar }(t^{\prime }-t) \hat{H} \right\} \hat{S}(t^{\prime},0) \\ && \times\exp \left\{ \frac{1}{i\hbar }(t^{\prime }-t)\hat{H}\right\} , \end{array} $$

(214)

and the initial-time condition at time t ₀, when the formalism begins to be applied, is

$$ \mathcal{R}_{\varepsilon }(t_{0})=\bar{\mathcal{R}}(t_{0},0). $$

(215)

The first time variables in the arguments of both $\bar {\mathcal {R}}$ and $\hat {S}$ refer to the evolution of the nonequilibrium-thermodynamic variables, and the second time variables, to the time evolution of the dynamical variables, both of which affect the operator.

The time t ₀, which starts the statistical description, is usually taken in the remote past ($t_{0}\rightarrow -\infty$), which introduces an adiabatic switching-on of the relaxation process. In (212), the time integration in the interval (t ₀,t) is weighted by the kernel $\exp \{ \varepsilon (t^{\prime }-t) \}$. The presence of this kernel introduces a kind of evanescent history as the system macrostate evolves toward the future from the boundary condition of (215) at time ($t_{0}\rightarrow -\infty$), a fact evidenced in the resulting kinetic theory [8–13, 16, 38], which clearly indicates that a fading memory of the dynamical process has been introduced. The statistical operator can be written in the form

$$ \mathcal{R}_{\varepsilon}(t)=\bar{\mathcal{R}}(t,0)+\mathcal{R}^{\prime} _{\varepsilon }(t)\;. $$

(216)

involving the auxiliary probability distribution $\bar {\mathcal {R}}(t,0)$, plus $\mathcal {R}^{\prime }_{\varepsilon }(t)$, which contains the historicity and irreversibility effects.

In most cases, we can consider the system as composed of the system of interest, i.e., the system on which we are performing an experiment, and ideal reservoirs in contact with it. Therefore, we can write the expression

$$ \bar{\mathcal{R}}(t,0) = \bar{\varrho}(t,0)\times \varrho_{\text{R}} \; . $$

(217)

and

$$ \mathcal{R}_{\varepsilon }(t)=\varrho_{\varepsilon }(t)\times \varrho_{\text{R}} \; , $$

(218)

where ϱ _ε (t) is the statistical operator of the nonequilibrium system, $\bar {\varrho }$ the auxiliary one, and ϱ _R the stationary operator for the ideal reservoirs, with ϱ _ε (t) then given by the equality

$$ \varrho_{\varepsilon }(t)=\exp \left\{ -\hat{S}(t,0)+\int\limits_{-\infty }^{t}dt^{\prime }e^{\varepsilon (t^{\prime }-t)}\frac{d}{dt^{\prime }}\hat{S}(t^{\prime },t^{\prime }-t)\right\} \, , $$

(219)

with the initial value $\bar {\varrho }(t_{0},0)$ ($t_{0}\rightarrow -\infty$), and where

$$ \hat{S}(t,0)=-\ln \bar{\varrho}(t,0) \;, $$

(220)

Finally, we have to provide the auxiliary statistical operator $\bar {\varrho }(t,0)$. This operator, sometimes dubbed the quasi-equilibrium statistical operator, expresses an instantaneous distribution at time t describing a “frozen” equilibrium that defines the macroscopic state of the system at that moment. On the basis of this notion (or, alternatively, via the variational procedure [12, 13, 96–104]) and considering the description of the system nonequilibrium state in terms of the basic set of dynamical variables $\hat {P}_{j}$, the reference or instantaneous quasi-equilibrium statistical operator is taken as a canonical-like one given by the equality

$$ \bar{\varrho}(t,0)=\exp \{-\phi (t)-{\sum\limits_{j}^{n}}F_{j}(t)\hat{P}_{j}\}\;, $$

(221)

with ϕ(t) ensuring the normalization of $\bar {\varrho }$ and playing the role of a logarithm of a partition function, say, $\phi (t)=\ln \bar {Z}(t)$.

Moreover, in (221), F _j is the nonequilibrium-thermodynamic variable associated with the basic dynamical variable $\hat {P}_{j}$. The nonequilibrium-thermodynamic space of states is composed by the basic variables {Q _j (t)}, which comprise the averages of the $\{ \hat {P}_{j} \}$ over the nonequilibrium ensemble, namely,

$$ Q_{j}(t) = \text{Tr}\{\hat{P}_{j}\varrho_{\varepsilon }(t)\}\,, $$

(222)

which are then functionals of the {F _j (t)} and there follow the equations of state

$$ Q_{j}(t) = - \frac{\delta \phi (t)}{\delta F_{j}(t)}=-\frac{\delta \ln \bar{Z}(t)}{\delta F_{j}(t)} \, , $$

(223)

where δ denotes functional derivative.

Moreover,

$$ \bar{S}(t) = \text{Tr}\{ \hat{\bar{S}}(t,0)\bar{\varrho}(t,0)\} = - \text{Tr}\{\bar{\varrho}(t,0)\ln \bar{\varrho}(t,0)\} \, , $$

(224)

is the so-called informational entropy characteristic of the distribution $\bar {\varrho }$, a functional of the basic variables {Q _j (t)}, in terms of which the following alternative form of the equations of state can be written:

$$ -\frac{\delta \bar{S}(t)}{\delta Q_{j}(t)}=F_{j}(t)\,, $$

(225)