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Mean field dynamics of a quantum tracer particle interacting with a boson gas
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Abstract
We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in . The pair interactions of the bosons are of mean-field type, with coupling strength proportional to where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit . Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.
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When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations, derived in \cite{tran2020boltzmann}.
We consider the dynamics of a large number N of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for general initial states, e.g., those that are expected to be ground states of interacting systems.
We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We prove that positive radial solutions of the quantum Boltzmann equation are bounded from below by a Gaussian, uniformly in time.
The Boltzmann–Nordheim equation is a modification of the Boltzmann equation, based on physical considerations, that describes the dynamics of the distribution of particles in a quantum gas composed of bosons or fermions. In this work we investigate the Cauchy theory of the spatially homogeneous Boltzmann–Nordheim equation for bosons, in dimension (Formula presented.). We show existence and uniqueness locally in time for any initial data in (Formula presented.) with finite mass and energy, for a suitable s, as well as the instantaneous creation of moments of all order.
The viscosity of a dilute Bose–Einstein condensate is obtained for a range of temperatures and densities. The kinetic equation used to derive the viscosity is based on Bogoliubov mean field theory and is a Boltzmann-like equation for the Bogoliubov excitations in the condensate. The viscosity can be assigned a numerical value for any gas for which the mass and scattering length are known. The viscosity decreases slowly as the temperature is decreased, but at low temperature (around 0.01 T c ), begins to show a much more rapid decrease with decreasing temperature. This interesting behavior of the viscosity is related to the behavior of the eigenvalues of the bogolon collision operator at these low temperatures.
This manuscript investigates the following aspects of the one dimensional
dissipative Boltzmann equation associated to variable hard-spheres kernel: (1)
we show the optimal cooling rate of the model by a careful study of the system
satisfied by the solution's moments, (2) give existence and uniqueness of
measure solutions, and (3) prove the existence of a non-trivial self-similar
profile, i.e. homogeneous cooling state, after appropriate scaling of the
equation. The latter issue is based on compactness tools in the set of Borel
measures. More specifically, we apply a dynamical fixed point theorem on a
suitable stable set, for the model dynamics, of Borel measures.
We consider the dynamics of N bosons in three dimensions. We assume that the pair interaction is given by . By studying an associated many-body wave operator, we introduce a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy
which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from
the beginning. Assuming energy conditions on the N-body wave function, for , we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the limit, all k-body scattering processes vanish if and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for and weakly for . Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we
have the estimates to achieve , it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space–time bound conjectured
by Klainerman and Machedon [48] for .
We construct infraparticle scattering states for Compton scattering in the standard model of non-relativistic QED. In our construction, an infrared cutoff initially introduced to regularize the model is removed completely. We rigorously establish the properties of infraparticle scattering theory predicted in the classic work of Bloch and Nordsieck from the 1930’s, Faddeev and Kulish, and others. Our results represent a basic step towards solving the infrared problem in (non-relativistic) QED. I.
We consider the 3D quantum BBGKY hierarchy which corresponds to the
N-particle Schr\"{o}dinger equation. We assume the pair interaction is
For interaction parameter , we prove that, as the limit points of
the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured
by Klainerman-Machedon in 2008. This allows for the application of the
Klainerman-Machedon uniqueness theorem, and hence implies that the limit is
uniquely determined as a tensor product of solutions to the Gross-Pitaevski
equation when the N-body initial data is factorized. The first result in this
direction in 3D was obtained by T. Chen and N. Pavlovi\'{c} (2011) for and subsequently by X. Chen (2012) for .
We build upon the approach of X. Chen but apply frequency localized
Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate
in the estimate of the potential part to extend the range to . Overall, this provides an alternative approach to the mean-field
program by Erd\"os-Schlein-Yau (2007), whose uniqueness proof is based upon
Feynman diagram combinatorics.
In this paper we provide a novel strategy to prove the validity of Hartree's
theory for the ground state energy of bosonic quantum systems in the mean-field
regime. For the well-known case of trapped Bose gases, this can be shown using
the strong quantum de Finetti theorem, which gives the structure of infinite
hierarchies of k-particles density matrices. Here we deal with the case where
some particles are allowed to escape to infinity, leading to a lack of
compactness. Our approach is based on two ingredients: (1) a weak version of
the quantum de Finetti theorem, and (2) geometric techniques for many-body
systems. Our strategy does not rely on any special property of the interaction
between the particles. In particular, our results cover those of Benguria-Lieb
and Lieb-Yau for, respectively, bosonic atoms and boson stars.
We study the dynamics of condensate formation in an inhomogeneous trapped Bose gas with a positive interatomic scattering length. We take into account both the nonequilibrium kinetics of the thermal cloud and the Hartree-Fock mean-field effects in the condensed and the noncondensed parts of the gas. Our growth equations are solved numerically by assuming that the thermal component behaves ergodically and that the condensate, treated within the Thomas-Fermi approximation, grows adiabatically. Our simulations are in good qualitative agreement with experiment, however important discrepancies concerning details of the growth behavior remain.
The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectra in weakly nonlinear and dispersive media. Such a kinetic model was derived by physicists in the 1960s, though the well-posedness theory remains open due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution-the first such result-to the wave turbulence equation for capillary waves.
We prove local existence and uniqueness results for the (space-homogeneous) 4-wave kinetic equation in wave turbulence theory. We consider collision operators defined by radial, but general dispersion relations satisfying suitable bounds, and we prove two local well-posedness theorems in nearly critical weighted spaces.
For the quantum kinetic system modelling the Bose-Einstein Condensate that accounts for interactions between condensate and excited atoms, we use the Chapman-Enskog expansion to derive its hydrodynamic approximations, include both Euler and Navier-Stokes approximations. The hydrodynamic approximations describe not only the macroscopic behavior of the BEC but also its coupling with the non-condensates, which agrees with Landau's two fluid theory.
This paper is devoted to the analysis of the continuous resonant (CR) equation, in dimensions greater than 2. This equation arises as the large box (or high frequency) limit of the nonlinear Schrodinger equation on the torus, and was derived in a companion paper by the same authors. We initiate the investigation of the structure of (CR), its local well-posedness, and the existence of stationary waves.
We consider the nonlinear Schr\"odinger (NLS) equation posed on the box with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in theory of statistical physics of dispersive waves, that goes by the name of "wave turbulence". Our main result is deriving a new equation, the continuous resonant (CR) equation, that describes the effective dynamics for large L and small over very large time-scales. Such time-scales are well beyond the (a) nonlinear time-scale of the equation, and (b) the Euclidean time-scale at which the effective dynamics are given by (NLS) on . The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.
We derive heuristically an integro-differential equation, as well as a shell
model, governing the dynamics of the Lowest Landau Level equation in a high
frequency regime.
We consider the continuous resonant (CR) system of the 2D cubic nonlinear
Schr{\"o}dinger (NLS) equation. This system arises in numerous instances as an
effective equation for the long-time dynamics of NLS in confined regimes (e.g.
on a compact domain or with a trapping potential). The system was derived and
studied from a deterministic viewpoint in several earlier works, which
uncovered many of its striking properties. This manuscript is devoted to a
probabilistic study of this system. Most notably, we construct global solutions
in negative Sobolev spaces, which leave Gibbs and white noise measures
invariant. Invariance of white noise measure seems particularly interesting in
view of the absence of similar results for NLS.
The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation.
We study the continuous resonant (CR) equation which was derived by
Faou-Germain-Hani as the large-box limit of the cubic nonlinear Schr{\"o}dinger
equation in the small nonlinearity (or small data) regime. We first show that
the system arises in another natural way, as it also corresponds to the
resonant cubic Hermite-Schr{\"o}dinger equation (NLS with harmonic trapping).
We then establish that the basis of special Hermite functions is well suited to
its analysis, and uncover more of the striking structure of the equation. We
study in particular the dynamics on a few invariant subspaces: eigenspaces of
the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space.
We focus on stationary waves and their stability.
This paper (published in 2013) introduces a refinement of the equations for
the pair excitation function used in our previous work with D. Margetis. The
new equations are Euler-Lagrange equations, and the solutions conserve energy
and the number of particles.
We consider an approximation of the linearised equation of the homogeneous
Boltzmann equation that describes the distribution of quasiparticles in a
dilute gas of bosons at low temperature. The corresponding collision frequency
is neither bounded from below nor from above. We prove the existence and
uniqueness of solutions satisfying the conservation of energy. We show that
these solutions converge to the corresponding stationary state, at an algebraic
rate as time tends to infinity.
This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead, we use the energy flow which is constant, but no more hydrodynamic.
We consider a non-relativistic quantum gas of N bosonic atoms confined to a
box of volume in physical space. The atoms interact with each other
through a pair potential whose strength is inversely proportional to the
density, , of the gas. We study the time evolution of
coherent excitations above the ground state of the gas in a regime of large
volume and small ratio . The initial state of
the gas is assumed to be close to a \textit{product state} of one-particle wave
functions that are approximately constant throughout the box. The initial
one-particle wave function of an excitation is assumed to have a compact
support independent of . We derive an effective non-linear equation
for the time evolution of the one-particle wave function of an excitation and
establish an explicit error bound tracking the accuracy of the effective
non-linear dynamics in terms of the ratio . We conclude
with a discussion of the dispersion law of low-energy excitations, recovering
Bogolyubov's well-known formula for the speed of sound in the gas, and a
dynamical instability for attractive two-body potentials.
Exact equations are obtained (in particular, for the anomalous self-energy part ..sigma../sub 12/(0) =0 and for the scattering vertex of two phonons GAMMA/sub 4/(p/sub i/..-->..0) =0), which indicate that in the general case the usual methods of summing the field perturbation-theory diagrams for Bose systems are not valid (the calculation must not be stopped when a converging result is obtained in the lower order in some small parameter). An investigation of the character of the infrared divergence of the field diagrams has yielded the region of applicability of the ordinary summation methods. It is shown that in the case T>0 or of a two-dimensional system at T=0 one must use a special regularization that calls for introducing the phonon vertices counterterms that, in contrast to the relativistic theories, contain no infinities and do not change the initial Hamiltonian at all. Examples of effective summation are presented for cases when the usual approach leads to an erroneous result (..sigma../sub 12/(p..-->..0), Pi (p..-->..0), and others). The derivation of the asymptotic Gavoret and Nozieres formulas for the Green's functions and the susceptibilities is re-examined with account taken of the equality ..sigma../sub 12/(0) =0.
We study the motion of a heavy tracer particle weakly coupled to a dense,
weakly interacting Bose gas exhibiting Bose-Einstein condensation. In the
so-called mean-field limit, the dynamics of this system approaches one
determined by nonlinear Hamiltonian evolution equations. We prove that if the
initial speed of the tracer particle is above the speed of sound in the Bose
gas, and for a suitable class of initial states of the Bose gas, the particle
decelerates due to emission of Cherenkov radiation of sound waves, and its
motion approaches a uniform motion at the speed of sound, as time tends to
infinity.
Preface; 1. Overview and introduction; 2. Condensate dynamics at T=0; 3.
Couple equations for the condensate and thermal cloud; 4. Green's
functions and self-energy approximations; 5. The Beliaev approximation
and the time-dependent HFB; 6. 6. Kadanoff-Baym derivation of the ZNG
equations; 7. Kinetic equations for Bogoliubov thermal excitations; 8.
Static thermal cloud approximation; 9. Vortices and vortex lattices at
finite temperatures; 10. Dynamics at finite temperatures using the
moment method; 11. Numerical simulation of the ZNG equations; 12.
Numerical simulation of collective modes at finite temperature; 13.
Landau damping in trapped Bose-condensed gases; 14. Landau's theory of
superfluidity; 15. Two-fluid hydrodynamics in a dilute Bose gas; 16.
Variational formulation of the Landau two-fluid equations; 17. The
Landau-Khalatnikov two-fluid equations; 18. Transport coefficients and
relaxation times; 19. General theory of damping of hydrodynamic modes;
Appendices; References; Index.
The paper considers the problem of the Bose-Einstein condensation in finite time for isotropic distributional solutions of the spatially homogeneous Boltzmann equation for Bose-Einstein particles with the hard sphere model. We prove that if the initial datum of a solution is a function which is singular enough near the origin (the zero-point of particle energy) but still Lebesgue integrable (so that there is no condensation at the initial time), then the condensation continuously starts to occur from the initial time to every later time. The proof is based on a convex positivity of the cubic collision integral and some properties of a certain Lebesgue derivatives of distributional solutions at the origin. As applications we also study a special type of solutions and present a relation between the conservation of mass and the condensation.
This paper deals with mathematical questions for Bose gases below the temperature T
BEC
where Bose-Einstein condensation sets in. The model considered is of two-component type, consisting of a kinetic equation for the distribution function of a gas of (quasi-)particles interacting with a Bose condensate, which is described by a Gross-Pitaevskii equation. Existence results and moment estimates are proved in the space-homogeneous, isotropic case.
We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two
dimensional box of size L with periodic boundary conditions. By taking the
large box limit in the weakly nonlinear regime (characterized by
smallness in the critical space), we derive a new equation set on that
approximates the dynamics of the frequency modes. This nonlinear equation turns
out to be Hamiltonian and enjoys interesting symmetries, such as its invariance
under Fourier transform, as well as several families of explicit solutions. A
large part of this work is devoted to a rigorous approximation result that
allows to project the long-time dynamics of the limit equation into that of the
cubic NLS equation on a box of finite size.
In this paper (the author’s Ph.D. thesis at ETH Zürich) the infrared problem is investigated in the framework of simple modeis (closely related to E. Nelson’s model [J. Math. Phys. 5, 1190–1197 (1964) doi:10.1063/1.1704225]). These modeis describe the interaction of conserved, charged, scalar particles (here called electrons) and relativistic, neutral, scalar bosons of restmass 0. They exhibit an “infra-particle Situation” in the sense of [B. Schroer, Fortschr. Phys. 11, 1–31 (1963)] and proper infrared problems in the construction of dressed one electron states and scattering amplitudes. A renormalized Hilbert space is constructed such that the spectrum of the energy-momentum operator on this Hilbert space contains a unique one electron shell corresponding to dressed one electron states. However, the spectrum of the energy-momentum operator on the physical Hilbert space does not contain a one electron shell. Several concepts for a collision theory on the charge one sector are developed. These concepts are compared with the proposals of L. D. Faddeev and P. P. Kulish [Theor. Math. Phys. 4, No. 2, 745–757 (1970); translation from Teor. Mat. Fiz. 4, No. 2, 153–170 (1970; Zbl 0197.26201)] and a list of interesting, yet unsolved problems is presented.
We present a new, simpler proof of the unconditional uniqueness of solutions
to the cubic Gross-Pitaevskii hierarchy in . One of the main tools in our
analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent
to the one established in the celebrated works of Erd\"os, Schlein and Yau,
\cite{esy1,esy2,esy3,esy4}.
The paper considers a model for Bose gases in the so-called ‘high-temperature range’ below the temperature where Bose-Einstein condensation sets in. The model is of non-linear two-component type, consisting of a kinetic equation with periodic boundary conditions for the distribution function of a gas of excitations interacting with a Bose condensate, which is described by a Gross-Pitaevskii equation. Results on well-posedness and long time behaviour are proved in a Sobolev space setting close to equilibrium.
The theory of gases in nonstationary states, given by Lorentz and Enskog, is generalized for the quantum statistics to give the hydrodynamical equations and the distribution function in first and second approximation, and formal expressions for the viscosity and heat conductivity coefficients. These results are valid for all statistics and for all degrees of degeneration. Two essential points contribute to this generality: (a) Exact expressions independent of statistics and degeneration can be given for the coordinate and time derivatives of the coefficient A of the equilibrium distribution function in terms of the pressure and temperature gradients and time derivatives, though a closed expression for this coefficient as a function of v and T is known only in limiting cases; (b) The function W of the general equation of state for ideal gases in all statistics pv=(RTM)W(v23T) is adiabatically invariant. Numerical values of the viscosity and heat conductivity coefficients, which should come out of the formal theory on the introduction of special assumptions about the molecular forces, have not yet been obtained. It is our hope that these results, when found, may furnish an experimental test of the existence of Einstein-Bose statistics in real gases, as is required by theory.
We consider the dynamics of a large system of N interacting bosons in the
mean-field regime where the interaction is of order 1/N. We prove that the
fluctuations around the nonlinear Hartree state are generated by an effective
quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's
approximation. We use a direct method in the N-particle space, which is
different from the one based on coherent states in Fock space.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
The theoretical description of trapped weakly interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the 'maze' of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain 'classes of approaches' identified by their physical content, rather than their particular mathematical implementation. This tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions, such as the occupation number representation of second quantization, are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean field, number conserving and stochastic) according to their particular needs. In addition to its 'training element', we hope that this tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to understand the approximate validity of these equations starting from a microscopic model. We discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process.
We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior. © 2015 by the American Mathematical Society. All rights reserved.
Expressions for the bulk viscosity, shear viscosity, and thermal conductivity of a quantum degenerate Bose gas above the critical temperature for Bose-Einstein condensation are derived using the Uehling-Uhlenbeck kinetic equation. For contact potentials and hard sphere interactions, the eigenvalues (relaxation rates) of the Uehling-Uhlenbeck collision operator have an upper cutoff. This cutoff requires summation over all discrete eigenvalues and eigenvectors of the collision operator when computing transport coefficients. We numerically compute the shear viscosity and thermal conductivity for any boson gas that interacts via a contact potential. We find that the bulk viscosity of the degenerate boson gas remains identically zero, as it is for the classical gas.
A functional integration method is used to obtain the first two terms of the asymptotic form of the Green's functions at (..omega..,k) =p..-->..0 and the principal asymptotic terms of the self-energy parts of three-dimensional and two-dimensional superfluid Bose systems at T=0. It is shown that the anomalous self-energy part tends to zero like (ln Rp)/sup -1/ for three-dimensional system and like p for the two-dimensional system.
We show how Boltzmann-Nordheim kinetic theory is modified by the existence of a Bose-
Einstein condensate. The final result is a set of coupled equations between the condensate
and the normal (thermal) component of the gas. This system shows a possible exchange of
mass between the condensate and the normal component through a kind of induced
emission preserving the coherence of the condensate.
We study the motion of a heavy tracer particle weakly coupled to a dense
interacting Bose gas exhibiting Bose-Einstein condensation. In the so-called
mean-field limit, the dynamics of this system approaches one determined by
nonlinear Hamiltonian evolution equations. We derive the effective dynamics of
the tracer particle, which is described by a non-linear integro-differential
equation with memory, and prove that if the initial speed of the tracer
particle is below the speed of sound in the Bose gas the motion of the particle
approaches an inertial motion at constant velocity at large times.
The Landau-Khalatnikov two-fluid hydrodynamic equations are derived for
a dilute, weakly interacting, condensed Bose gas on the basis of a
microscopic theory. Explicit expressions for the transport coefficients
in the linearized equations are given for very low temperatures and for
moderately low temperatures below the λ point.
This comprehensive two-volume introduction to a modern and rapidly
developing field starts at the level of graduates and young researchers.
It provides a general theory of developed turbulence with a consistent
description of phenomena in different media such as plasmas, solids,
atmosphere, oceans and space. This volume starts with simple dimensional
analysis and proceeds to rigorous theory with exact solutions for the
stationary spectra of turbulence, the solution of the stability problem,
matching of Kolmogorov-like spectra with pumping and damping. The reader
is provided with the necessary tools for studying nonlinear waves and
turbulence: Hamiltonian formalisms, methods of statistical description,
derivation of kinetic equations and their steady and nonsteady
solutions.