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On the deduction of Markoffian master equation
Abstract
The solution of the generalized master equation of Zwanzig for a macroscopic system, is approximated by the solution of a markoffian master equation. The reliability of this approximation is studied at arbitrary times.
The part Φ0(t) of the statistical operator (or density function) which is relevant for the description of macroscopic dynamics is treated. The new mathematical properties of the solution of Zwanzig's generalized master equation which are important for the deduction of a markoffian master equation for Φ0(t) are pointed out. On the basis of such results the conditions under which markoffian macroscopic dynamics exists are discussed. A comparison with the results of I. Prigogine's school is finally made.
Mathematical aspects of the application of the projection-operator method for the derivation of the generalized master or kinetic equations are discussed. A new form of the convolutionless equation is derived using the method of time-dependent projection operators. Some examples of the application of the method are given. In particular the generalized Boltzmann kinetic equation and the master equation for the N-particle velocity distribution are derived. The long-time asymptotic form of the scattering operator of the master equation is considered and shown to be identical with the results of the theory of Prigogine et al.
To give a macroscopic description of a system, a suitable assumption is generally made on the initial statistical operator. We look for the situation in which the Liouville-von Neumann equation implies that an initial condition of such type can be again assumed at time t1, once it has been assumed at some time t0< t1. It is pointed out that, on the contrary, such initial condition cannot be assumed at time t1, after a time-reflection has been made. Such lack of symmetry is very relevant for the understanding of macroscopic irreversibility.
A general theorem is deduced which illuminates the problem of extracting reduced dynamics from the general dynamics of a system. The Liouville-space formalism is used throughout. The results are applied to the problem of deducing a self-contained description for the time evolution of the macroscopic observables of a microscopic system.By our treatment it emerges clearly why a markoffian master equation holds in general only approximately; further, a possible generalization of quantum mechanics is indicated, by which an independent dynamics of macroscopic observables holds exactly.
Unlike the alkali halides with NaCl structure, the CsCl-structure alkali halides present several problems regarding the cohesive energy, the static lattice structure, the dielectric properties and the phonon dispersion relation. From a critical discussion of the problems it is seen that these differences from the rest of the alkali halides become all the more prominent, when any attempt is made to reproduce the above properties simultaneously in any of the existing models. An attempt is made to calculate simultaneously the lattice statics and dynamics of the CsCl and CsI crystals from a single model with a single set of model parameters. The results of the calculation show that a consistent description for most of the major properties is obtained in the model. The source of the remaining discrepancy is discussed.
Measurements between 90K and 200K of the field outside the domain, the domain velocity, and other entities associated with slow domains in n-type GaAs:O (2 Omega cm at 300K) are shown to point unambiguously to the origin of the associated negative differential resistance. This origin is an impurity-barrier mechanism connected with trapping at Gaussian-broadened complex of levels peaking near 0.05 eV below the conduction band. The natures of the barrier and centre are unknown. It is shown that if the barrier is taken to be unscreened Coulombic originating in a negative charge of magnitude Ze then the measured activation energy (0.077 eV) entails Z=10, which appears unacceptably high.
The authors discuss the nature of deformation of the electron charge cloud of an ion implied in the deformable shell model and considers an application of the model to the case of LiF crystals. It is found that the properties of the crystal are fairly well reproduced by the present model and certain discrepancies noted by earlier authors are removed. The results of other theoretical investigations are also included for comparison.
Slow domains in oxygen doped n-type GaAs with photoexcited conductivities ranging from 10-8 to 10-4 ( Omega CM)-1 have been investigated in the temperature range 100 to 200 K. It is proposed that the origin of these domains is the transferred electron effect with enhanced trapping from the (100) valley. The low threshold voltages are attributed to high resistance layers near the alloyed contacts of width about 10 mu m in low resistivity samples and about 100 mu m in high resistivity samples. A theoretical model is proposed for the domain incubation which invokes a sub nl product behaviour of the cathode layer and compared with experimental results of incubation time against applied field and the variation of threshold field with sample length. The trap-modified drift velocity against field curve has been calculated from the results of Fawcett et al (1970) for the transferred electron effect in pure GaAs at 300K. The ratio of the capture time in the (000) valley to that in the '100) valley is estimated to be about thirty from the measured extra-domain drift velocity, which was 4*105cm s-1 at temperatures around 150 K.
Starting from the Liouville equation and making use of projection operator techniques we obtain a compact equation for the
rate of change of then-particle momentum distribution function to any order in the density. This equation is exact in the thermodynamic limit. The
terms up to second order in the density are studied and expressions are given for the errors committed when one makes the
usual hypothesis to derive generalized Boltzmann equations. Finally the Choh-Uhlenbeck operator is obtained under additional
assumptions.
Partendo dall'equazione di Lioville e facendo uso della tecnica degli operatori di proiezione, si ottiene un'equazione compatta
per il rapporto di variazione della funzione di distribuzione dell'impulso din particelle per qualsiasi ordine della densità. Questa equazione è esatta nel limite termodinamico. Nel dedurla si suppone
solo che il gas sia inizialmente uniforme e che le correlazioni siano inizialmente di rango finito. Si studiano i termini
sino al secondo ordine nella densità e si danno espressioni per gli errori che si commettono quando si fa l'usuale ipotesi
per dedurre le equazioni di Boltzmann generalizzate. Infine per mezzo di ipotesi aggiuntive si ottiene l'operatore di Choh-Uhlenbeck.
Исходя из уравнения Лиувилля и используя технику операторов проектирования, мы получаем компактное уравнение для интенсивности
изменения функцииn-частичного импульсното распределения в любом порядке по плотности. Это уравнение является точным в термодинамическом пределе.
Исследуются члены вплоть до второго порядка по плотности и приводятся выражения для допущенных погрешностей, когда используются
обычные гипотезы при выводе обобщенных уравнений Больцмана. В эаключение получается оператор Коха-Ухленбека при дополнительных
предположениях.
Some unpleasant features of the usual treatment of irreversible processes in quantum mechanics are discussed. It is shown how the description of a non-relativistic unstable particle can be cleanly embedded into a reversible Galilean quantum field theory. It is proven that in the case of stable particles the embedding procedure gives the same values for internal energy which are obtained by the usual procedure. Finally the technique is applied to the Galilee model.
A contribution is given to the attempts towards a solution of the measurement problem in quantum mechanics, in the spirit
of some previous papers. In order to justify the possibility of a macroscopic description of the measuring apparatus, a relevant
hypothesis is found to be the separation of two characteristic times: the decay time for the kernel of a generalized master
equation and the characteristic time for the evolution of macroscopic quantities of the measuring device.
Si fornisce un contributo alla ricerca di una soluzione del problema quantistico della misurazione, seguendo la linea di precedenti
lavori. Si trova che, nel giustificare la possibilità di descrivere macroscopicamente l'apparecchio di misura, gioca un ruolo
importante la separazione di due tempi caratteristici: l'uno, tempo caratteristico per il decadimento del nucleo di una master
equation generalizzata, l'altro, tempo caratteristico per l'evoluzione delle grandezze macroscopiche relative all'apparecchio
di misura.
Exact equations of motion for a system interacting with a reservoir are derived from the exact evolution law of the composed system by means of projection-operator techniques. The following two cases are considered: a) an arbitrary system, b) a system consisting of weakly interacting particles.The equation of motion obtained in case b) is used as a starting point for the derivation of a Pauli master equation for a system consisting of weakly interacting particles weakly coupled to a reservoir and the range of validity of this markovian law is discussed extensively.The kernel of the master equation obtained in case a) is investigated in the thermodynamic limit of the reservoir. A diagonal singularity analogous to that of Van Hove is used. The problem of ergodicity and the derivation of a markovian master equation to general order in the system-reservoir interaction are discussed in brief.
The following considerations shall contribute to our understanding of the macroscopic-stochastic and macroscopic-deterministic properties of quantum gases. We shall begin with a physical formulation of the problem, which resembles Boltzmann's point of view in classical mechanics. Just the difference to Gibb's method, to which in quantum mechanics corresponds the discussion of the Wigner distribution function, is important to understand the properties of the many body system as seen by a macroscopic observer.
We use the following method: Taking localized one particle functions (localized in configuration space as well as in momentum space) we define particle number operators, whose proper values or expectation values are the Boltzmann distribution function of the considered quantum gas, if measured in suitable “macroscopical ensembles”. In order to discuss its time development the Hamiltonian of the gas is translated into the occupation number representation with these localized wave functions. The result is: There is not only a change of the distribution function due to the macroscopically wellknown effects of streaming und collisions, but there would also be typical microscopical quantum effects, if the one particle functions are not chosen in a certain form together with a restriction in the desired observation. Thus the possible macroscopic observations in quantum gases are marked out. Our considerations also give a concrete example for the general concept of macroscopic observables.
Prigogine and his co-workers developed a genera; theory of approach to
equilibrium in classical mechanics. A inethod is presented which indicates the
possibility for obtaining similar results for quantum systems. This method makes
use of a representation for the density matrix which stresses the formal analogy
between the von Neumann equation and the classical Liouville equation. This
representation was used in the particular cases of weakly-coupled systems and of
quantum plasmas. (B.O.G.)
A new generalization of Onsager's theory of irreversible processes is presented. The main purpose is to allow for memory effects or causal time behavior, so that the response to a thermodynamic force comes later than the application of the force. This is accomplished by a statistical mechanical derivation of an exact non-Markoffian kinetic equation for the probability distribution in the space of macroscopic state variables. The memory effect in the resulting transport equations is represented by a time convolution of the thermodynamic forces with memory functions. The latter are time-correlation functions in the rates of change of the phase functions corresponding to macroscopic quantities. The resulting transport equations are not restricted to small deviations from thermal equilibrium. Onsager's theory is shown to be the low-frequency limit of our causal theory.
Nous appliquons la technique de diagrammes développée par l'un de nous (I.P.) et R. Balescu, combinée au formalisme de la résolvante, pour établir les équations générales d'évolution d'un gaz classique à partir de l'équation de Liouville.
Nous obtenous des équations valables à tous les ordres dans le paramétre de couplage et dans la concentration, et pour tous les temps. Pour des temps courts, ces équations sont non-Markoviennes et tiennent compte de la durée finie d'une collision; dans la limite des temps longs (t→∞), elles se ramènent aux expressions asymptotiques Markoviennes obtenues précédemment dans l'étude de l'approche vers l'équilibre.
Nous discutons rapidement le lien entre cette théorie et le schéma proposé par Bogolioubov. Enfin, nous montrons que, pour le calcul des coefficients de transport le caractère non Markovien de la cinétique de l'approche vers l'équilibre ne joue pas de rôle: ces coefficients sont entièrement déterminés par les sections efficaces asymptotiques.
Nous appliquons la mthode gnrale dveloppe dans Particle prcdent h un certain nombre de situations simples. Dans le cas des gaz homognes faiblement coupls nous redrivons les quations obtenues prcdemment par l'un de nous (I.P.) et ses collaborateurs. Nous pouvons cependant dduire systmatiquement par notre mthode les quations d'volution aux approximations suivantes en (le paramtre de couplage). L'tude des gaz inhomognes nous conduit en premire approximation une quation de Boltzmann, alors que l'approximation suivante introduit un champ moyen du type tudi par V 1 a s s o v. Nous traitons aussi brivement le cas des gaz dilus interactions fortes. Enfin nous discutons la signification et l'origine de l'irreversibilit des systmes mcaniques ayant un grand nombre de degrs de libert.
We describe a new formulation of methods introduced in the theory of irreversibility by Van Hove and Prigogine, with the purpose of making their ideas easier to understand and to apply. The main tool in this reformulation is the use of projection operators in the Hilbert space of Gibbsian ensemble densities. Projection operators are used to separate an ensemble density into a ``relevant'' part, needed for the calculation of mean values of specified observables, and the remaining ``irrelevant'' part. The relevant part is shown to satisfy a kinetic equation which is a generalization of Van Hove's ``master equation to general order.'' Diagram summation methods are not used. The formalism is illustrated by a new derivation of the Prigogine‐Brout master equation for a classical weakly interacting system.
The dynamics of a set of macroscopic variables of a system is formulated by means of a quantum-statistical treatment that provides microscopic conditions for which the variables evolve according to a closed phenomenological law. The theory follows an approach delineated in an earlier paper1), in which an exact generalised master equation for microscopic variables was first reduced to a Markoffian form and then treated so as to yield deterministic phenomenological equations. This approach is now systematised by a reformulation that provides conditions for the validity of the procedure from a generalised master equation to a deterministic law.Thus the macroscopic observables are now formulated in a scheme where some of their characteristic properties, stemming from their many-particle structures, are incorporated into the general ‘phase cell’ representation of Van Kampen2) and Emch3). These properties are expressed in terms of parameters representing characteristic values of the variables of the theory. Of particular importance is a dimensionless size parameter, Γ, defined as a characteristics ratio of a microscopic variable to a microscopic quantum.the incorporation of these characteristic properties into Emch's3) generalised master equation yields conditions under which a specified set of macroscopic variables conforms to a closed phenomenological law, to lowest order in Γ-1. Higher order corrections represent ‘memory effects’ and fluctuations.
A master equation governing the dynamical evolution of the macroscopic state of a closed system is derived from the microscopic equations of motion by a treatment that involves no assumption of ‘repeated random phases’. Phenomenological equations, comprising nonlinear relations between fluxes and driving forces, are derived from the master equation. The transport coefficients that appear in these latter equations are formulated in terms of microscopic properties of the system.The theory yields generalisations of Onsager's linear phenomenological theory and of the Kubo formalism for transport coefficients in thermally-driven systems.
The approach of a quantum system to statistical equilibrium under the influence of a perturbation is described by a well known transport equation, now often called master equation (see (1.1) hereunder). This equation holds only when the perturbation is taken into account to lowest non-vanishing order. It has been stressed in a recent paper that certain characteristic properties of the perturbation, easily seen to hold for actual systems (crystals, gases), play an essential role in determining the irreversible nature of the effects described by (1.1). On the basis of these properties it was possible to derive the lowest order master equation from the Schrödinger equation by making one assumption only, relative to the phases of the wave function at the initial time. In contrast with the usual derivation which assumes the phases to be random at all times, the method just mentioned is capable of extension to higher orders in the perturbation. This extension is carried out in the present paper. The essential results are the establishment of a generalized master equation valid to arbitrary order in the perturbation, and the proof that the long time behaviour of its solution corresponds to establishment of microcanonical equilibrium (the latter being taken for the total hamiltonian, perturbation included). The generalized master equation exhibits with its lowest order version the essential difference that it corresponds to a non-markovian process. The transition from the exact master equation to its lowest order approximation is discussed in detail. It illustrates the existence of two time scales, a short one and a long one, for very slow irreversible processes, as well as their overlapping in the case of faster processes.