THE JOURNAL OF CHEMICAL PHYSICS 140, 034111 (2014)
Nonlinear response theory in chemical kineticsMaksym Kryvohuz1,a) and Shaul Mukamel2,b)1Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA2Chemistry Department, University of California, Irvine, California 92697-2025, USA
(Received 19 September 2013; accepted 23 December 2013; published online 17 January 2014)
A theory of nonlinear response of chemical kinetics, in which multiple perturbations are used toprobe the time evolution of nonlinear chemical systems, is developed. Expressions for nonlinearchemical response functions and susceptibilities, which can serve as multidimensional measures ofthe kinetic pathways and rates, are derived. A new class of multidimensional measures that combinemultiple perturbations and measurements is also introduced. Nonlinear fluctuation-dissipation rela-tions for steady-state chemical systems, which replace operations of concentration measurement andperturbations, are proposed. Several applications to the analysis of complex reaction mechanisms areprovided. 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861588]
Great variety of problems in chemistry and biology in-volve complex reaction mechanisms.1, 2 These include elabo-rate reaction networks in combustion and atmospheric chem-istry, complex enzyme catalyzed reactions in biology, proteinnetworks in living cells and others. Analysis of the topologyof complex reaction networks and their elementary steps re-quire development of suitable experimental techniques.
In this paper we propose multidimensional time-dependent techniques, drawing upon the analogy with time-resolved nonlinear spectroscopy. In such experiments, the dy-namical system of interest is subjected to a series of controlledexternal perturbations and its response is measured at a latertime. The time intervals between the perturbations and themeasurement constitute a multidimensional parameter space,which caries detailed information about the system. Multidi-mensional techniques are not normally applied for analysis ofchemical kinetics, and this paper aims at extending the appli-cability of multidimensional methods of response to perturba-tions into the field of chemical kinetics.
Linear response experiments have been widely employedto study complex reaction mechanisms.311 Nonlinear re-sponse of chemical systems have been also addressed,1216
mostly for one-dimensional, i.e., single-time nonlinear exper-iments, without resorting to multidimensional response sig-nals. However, nonlinear signal of nth order implies responseof a chemical system to n external perturbations, which canbe applied and measured in n time intervals, and thus gener-ally form an n-dimensional characterization of the chemicalsystem. The nonlinearity comes from the kinetic equationsdescribing time evolution of a chemical system, e.g., bimolec-ular reactions, catalyzed reactions, etc.
In the present paper, we show that nonlinear responsefunctions formally similar to those of classical systems inphase space can be derived with minor modification for gen-eral nonlinear dynamical systems such as chemical reactions.
a)Electronic mail: [email protected])Electronic mail: [email protected]
The canonical phase space structure is not essential. Similar toclassical response theory in phase space, expressions of non-linear chemical response functions contain stability deriva-tives, which potentially carry detailed dynamical information.We further show that fluctuation dissipation theorems can bederived for chemical systems out of equilibrium, e.g., steadystates, provided they satisfy gaussian statistics.
The paper is organized as follows. In Sec. II, we presentformal perturbative expressions for the nonlinear response ofchemical systems. In Sec. III, we illustrate that the resultingperturbative series reproduce some exactly solvable problemsin chemical kinetics. In Sec. IV, we consider several specialforms of perturbations of chemical systems and obtain finitedifference expressions for nonlinear response functions andsusceptibilities. In Sec. V, we extend the class of nonlinearexperiments to multiple perturbations and multiple measure-ments, and show that they carry complementary informationto the nonlinear response functions. In Sec. VI, we deriveapproximate steady-state fluctuation-dissipation relationsbetween the ordinary and generalized nonlinear responsefunctions. In Sec. VII, we give several examples of the newinformation provided by multidimensional signals about themechanisms of chemical reactions. The paper is concludedwith discussion in Sec. VIII.
II. NONLINEAR RESPONSE OF A GENERALDYNAMICAL SYSTEM
The time evolution of a general dynamical system can bedescribed via a set of first order differential equations
dt= hi(x, t), (2.1)
where xi stands for the ith component of the N-dimensionalvector x of dynamical variables. If higher order time deriva-tives are involved in the equations of motion, they still canbe represented in the form of Eq. (2.1) by replacing higherorder derivatives with new dynamical variables. The ex-plicit time dependence of hi in Eq. (2.1) implies variation of
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034111-2 M. Kryvohuz and S. Mukamel J. Chem. Phys. 140, 034111 (2014)
systems parameters with time. Examples of dynamical sys-tems that can be described via Eq. (2.1) include:
(1) Chemical reactions,
kj (t)Cn11 . . . C
nNN . (2.2)
(2) Classical Hamiltonian systems in phase space,
dt= V (x(t))
(3) Currents in electric circuits,
where R depends nonlinearly on I, and many others. Whilein this paper we will be mostly concerned with chemical sys-tems, the results can be applied to a broader range of systems,whose time evolution is of the form in Eq. (2.1).
One of the most commonly used experimental ap-proaches to study complex dynamical systems is to measuretheir response to a controlled external perturbation. For phys-ical systems such perturbation could correspond to an ex-ternally applied force (such as laser pulse in optical spec-troscopy), while for chemical systems it can correspond toa controlled change in concentration of a particular chemi-cal species. Denoting a generalized external perturbation byhi(x, t), the equations of motion (2.1) become
dt= hi(x, t) + hi(x, t), (2.5)
where hi(x, t) determines the unperturbed systems dynam-ics, usually at equilibrium or in a stationary state.
The measured quantities, f (x), i.e., the observables, aresome functions of the quantities being perturbed, xi. Math-ematically, we are interested in expressing the response ofthe observable f (x) in terms of the imposed perturbationhi(x, t). This can be efficiently done using time dependentperturbation theory.1719 Since most derivations of time de-pendent perturbation theory are done for Hamiltonian sys-tems, it will be instructive to provide a derivation for thegeneral (non-Hamiltonian) dynamical system of the formin Eq. (2.5).
Let us consider an observable f (x) which is an arbi-trary differentiable function of coordinates x. Since df (x)/dt= j (f (x)/xj )(dxj (t)/dt), the evolution of f (x(t)) isgiven by
= (D0(t) + D(t)) f (x), (2.6)
where the differential operators D0(t) and D(t) are
Here D(t) represents the combined effect of all perturbationsat time t on the observable f. D(t) vanishes as the strengthof perturbations hi goes to zero. We assume that initially thesystem is unperturbed, hi|t = 0 = 0, and thus D(0) = 0.
Equation (2.6) can be solved iteratively and its solutioncan be expressed in the known form19
f (x(t)) = D(t)f (x)|x=x(0) , (2.8)where we have introduced evolution (time translation) opera-tor D(t),
D(t) = T exp t
= 1 + t
0D( )d +