Statistical Thermodynamics of the FröhlichBoseEinstein Condensation of Magnons out of Equilibrium
Abstract
A nonequilibrium statisticalthermodynamic approach to the study of a FröhlichBoseEinstein condensation of magnons under radiofrequency radiation pumping is presented. Such a system displays a complex behavior consisting in steadystate conditions to the emergence of a synergetic dissipative structure resembling the BoseEinstein condensation of systems in equilibium. A kind of “two fluid model” arises: the “normal” nonequilibrium structure and Fröhlich condensate or “nonequilibrium” one, which is shown to be an attractor to the system. We analyze some aspects of the irreversible thermodynamics of this dissipative complex system, namely, its informational entropy, expressions for the fluctuations in nonequilibrium conditions, the associated Maxwell relations and the formulation of a generalized theorem. We also study the informational entropy production of the system, an order parameter is introduced and GlansdorffPrigogine criteria for evolution and (in)stability are verified.
I Introduction
It has been noticed kruglyak2010 () that the study of collective spin excitations in magnetically ordered materials (socalled spin waves and the associated quasiparticles, the magnons) has a successful history of more than 80 years bloch1930 (), which recently has reemerged within a young field of research and technology referredto as Magnonics. This term Magnonics is considered to describe the subfield of magnetic dynamic phenomena. The name Magnonics was created by analogy with Electronics, with the magnons acting in the transference of information instead of the electron charges in devices.
One important result pertaining to Magnonics has been the observation of a macroscopic quantum phenomenon resembling a BoseEinstein condensation of magnons excited out of equilibrium by action of an electromagnetic field in the radiofrequency portion of the spectrum.
The kinetic of evolution of the system of spins in thin films of yttriumirongarnets (YIG) in the presence of a constant magnetic field, and being excited by a source of rfradiation which drives the system towards farremoved from equilibrium conditions, has been reported in detailed experiments performed by Demokritov et al. demokritov2006 (); demidov2008 (). These experimental results have evidenced the occurrence of an unexpected large enhancement of the population of the magnons in the state lowest in energy in their energy dispersion relation. That is, the energy pumped on the system instead of being redistributed among the magnons in such nonthermal conditions is transferred to the mode lowest in frequency (with a fraction of course being dissipated to the surrounding media). Some theoretical studies along certain approaches has been presented by several authors (see for example Refs. tupitsyn2008 (); rezende2009 (); malomed2010 ()); we proceed here to describe the phenomenon within a complete thermostatistical description within the framework of a nonequilibrium ensemble formalism.
Such phenomenon has been referredto as a nonequilibrium BoseEinstein condensation, which then would belong to a family of three types of BEC:
The original one is the BEC in manyboson particle systems in equilibrium at very low temperatures, which follows when their de Broglie thermal wave length becomes larger than their mean separation distance, and presenting some typical hallmarks (spontaneous symmetry breaking, longrange coherence, etc.). Aside from the case of superfluidity, BEC was realized in systems consisting of atomic alkali gases contained in traps. A nice tutorial review is due to A. J. Leggett leggett2001 () (see also pitaevskii2003 ()).
A second type of BEC is the one of bosonlike quasiparticles, that is, those associated to elementary excitations in solids (e.g. phonons, excitons, hybrid excitations, etc.), when in equilibrium at extremely low temperatures. A well studied case is the one of an excitonpolariton system confined in microcavities (a near twodimensional sheet), exhibiting the classic hallmarks of a BEC snoke2010 ().
The third type, the one we are considering here, is the case of bosonlike quasiparticles (associated to elementary excitations in solids) which are driven out of equilibrium by external perturbative sources. D. Snoke snoke2 () has properly noticed that the name BEC can be misleading (some authors call it “resonance”, e.g. in the case of phonons phonon ()), and following this author it is better not to be haggling about names, and we introduce the nomenclature NEFBEC (short for NonEquilibrium FröhlichBoseEinstein Condensation for the reasons stated below). As noticed, here we consider the case of magnons (bosonlike quasiparticles), demonstrating that NEFBEC of magnons is another example of a phenomenon common to manyboson systems embedded in a thermal bath (in the conditions that the interaction of both generates nonlinear processes) when driven sufficiently away from equilibrium by the action of an external pumping source and which display possible applications in the technologies of devices and medicine.

A first case was evidenced by Herbert Fröhlich who considered the many boson system consisting of polar vibration (lo phonons) in biopolymers under dark excitation (metabolic energy pumping) and embedded in a surrounding fluidfroehlich1970 (); froehlich1980 (); mesquita1993 (); fonseca2000 (). From a Science, Technology and Innovation (STI) point of view it was considered to have implications in medical diagnosishyland1998 (). More recently has been considered to be related to brain functioning and artificial intelligencepenrose1994 ().

A second case is the one of acoustic vibration (ac phonons) in biological fluids, involving nonlinear anharmonic interactions and in the presence of pumping sonic waves, with eventual STI relevance in supersonic treatments and imaging in medicinelu1994 (); mesquita1998 ().

A third one is that of excitons (electronhole pairs in semiconductors) interacting with the lattice vibrations and under the action of rfelectromagnetic fields; on a STI aspect, the phenomenon has been considered for allowing a possible excitonlaser in the THz frequency range called “Excitoner”mysyrowics1996 (); mesquita2000 ().

A fourth one is the case of magnons already referred to demokritov2006 (); demidov2008 (), which we here analyze in depth. The thermal bath is constituted by the phonon system, with which a nonlinear interaction exists, and the magnons are driven arbitrarily out of equilibrium by a source of electromagnetic radio frequency vannucchi2010 (); vannucchi2013 (). Technological applications are related to the construction of sources of coherent microwave radiation demodov2011 (); ma2011 ().
There exist two other cases of NEBEC (differing from NEFBEC) where the phenomenon is associated to the action of the pumping procedure of drifting electron excitation, namely,

A fifth one consists in a system of longitudinal acoustic phonons driven away from equilibrium by means of drifting electron excitation (presence of an electric field producing an electron current), which has been related to the creation of the socalled Saser, an acoustic laser device, with applications in computing and imaging kent2006 (); rodrigues2011 ().

A sixth one involving a system of LOphonons driven away from equilibrium by means of drifting electron excitation, which displays a condensation in an offcenter small region of the Brillouin Zone rodrigues2010 (); komirenko2000 ().
We describe here item number 4, namely, a system of magnons excited by an external pumping source. For that purpose, we consider a system of localized spins in the presence of a constant magnetic field, being pumped by a rfsource of radiation driving them out of equilibrium while embedded in a thermal bath consisting of the phonon system (the lattice vibrations) which is considered to be in equilibrium with an external reservoir at temperature . The microscopic state of the system is characterized by the full Hamiltonian of spins and lattice vibrations after going through HolsteinPrimakov and Bogoliubov transformationskeffer1966 (); akhiezer1968 (); white1983 (). On the other hand, the characterization of the macroscopic state of the magnon system is done in terms of the ThermoMechanical Statistics based on the framework of a NonEquilibrium Statistical Ensemble Formalism (NESEF for short)luzzilivro2002 (); luzzi2006 (); zubarev1996 (); kuzensky2009 (); akhiezer1981 (); mclennan1963 (). Other modern approach consists in the use of Computational Modelingkalos2007 (); frenkel2002 () (developed after Nonequilibrium Molecular Dynamicsalder1987 ()). It may be noticed that NESEF is a systematization and an extension of the essential contributions of several renowned authors following the brilliant pioneering work of Ludwig Boltzmann. The formalism introduces the fundamental properties of historicity and irreversibility in the evolution of the nonequilibrium system where dissipative and pumping processes are under way.
In terms of the dynamics generated by the full Hamiltonian the equations of evolution of the macroscopic state of the system are obtained in the framework of the NESEFbased nonlinear quantum kinetic theory luzzilivro2002 (); luzzi2006 (); zubarev1996 (); kuzensky2009 (); akhiezer1981 (); mclennan1963 (); lauck1990 (); kuzensky2007 (); madureira1998 (); vannucchi2009osc (). We call the attention to the fact that the evolution equations are the quantum mechanical equations of motion averaged over the nonequilibrium ensemble, with the NESEFkinetic theory providing a practical way of calculation. The evolution of the nonequilibrium state of magnons under rfradiation excitation is fully described in Refs. vannucchi2010 (); vannucchi2013 () (for the sake of completeness we summarize the results in Section II), and, on the basis of it we present here a extended study of the nonequilibrium irreversible thermodynamics of the FröhlichBoseEinstein condensation of such “hot” magnons. This is done in terms of the NESEFbased NonequilibriumStatistical Irreversible Thermodynamics luzzi2000 (); luzzi2001 () (also Ch. 7 in Ref. luzzilivro2002 ()).
Ii FröhlichBoseEinstein Condensation of hot magnons in briefvannucchi2010 (); vannucchi2013 ()
The system we are considering consists of a subsystem of spins being pumped by a microwave source and interacting nonlinearly with a thermal bath (blackbody radiation and crystalline lattice) that is in contact with a thermal reservoir in equilibrium at temperature . This system is well described by the Hamiltonian
(1) 
where accounts for the internal (exchange and magnetic dipole) interactions between spins, is associated with the effect of the constant magnetic field (Zeeman Effect). and are the Hamiltonian of the thermal bath (lattice and radiation respectively), and their interaction with the spin subsystem ( includes also the effect of the source). Introducing the quasiparticles related to the spin, lattice and radiation variables (respectively the magnons, phonons and photons) and their creation and annihilation operators (, , , , and ) we may write the Hamiltonian of Eq. (1) as
(2) 
with
(3) 
being the noninteracting term formed by the Hamiltonians of free magnons, phonons and photons, and , and their energies. The other term,
includes the interactions between quasiparticles:
(4) 
is the magnonmagnon scattering term;
(5) 
accounts for the relevant magnonphonon interaction and
(6) 
is the interaction between magnons and photons (source and blackbody radiation).
After the mechanical description of the system follows the thermodynamical one. The thermodynamical state can be defined in terms of the timedependent thermodynamical variables
(7) 
average values of the socalled basic microdynamical variables
(8) 
with , where
(9) 
is the hamiltonian of the thermal bath (phonons and photons),
(10) 
with being the population operator of magnons in mode , describing its change in space (inhomogeneities in populations) and we recall that () are singlemagnons operators whose eigenstates are the coherent states. Finally,
(11) 
are the HugenholtzGorkov pairs of two magnons.
These averages are weighted through a nonequilibrium statistical operator , for example, . We introduce a factorization between the thermal bath in equilibrium and the magnetic subsystem
(12) 
where
(13) 
is the canonical distribution function of the phonons and photons in stationary condition near equilibrium at temperature (being its partition function) and is the nonequilibrium statistical operator of the magnon system. The last one may be obtained solving a modified LiouvilleDirac equation for ,
(14) 
where the right term (with ) introduces the “Bogoliubov’s symmetrybreaking procedure” in time and is the auxiliary statistical operator. Equation (14) ensures on the one hand that the nonequilibrium statistical operator incorporates the dynamical evolution while, on the other hand, includes irreversibility luzzilivro2002 (); luzzi2006 (); zubarev1996 ().
The auxiliary statistical operator is written in terms of the chosen microdynamical variables taken the form
(15) 
where
(16) 
are the nonequilibrium thermodynamic variables conjugated to the basic variables contained in set (7) in the sense of the Eqs. (18) and (19) below. The normalization of introduces the nonequilibrium partition function
(17) 
It is important here to make three observations: first, we stress that the auxiliary statistical operator does not describe the irreversible timeevolution of the system, and the average values weighted with coincide with those weighted with only for the microdynamical variables, for example . Second, the expression adopted in Eq. (15) for the statistical operator has the form of an instantaneous generalized canonical distribution that tends to the canonical one when the system is in equilibrium with all the present intensive variables except (that, in this case, is associated with the magnons equilibrium temperature) going to zero. Finally, since the intensive nonequilibrium thermodynamic variables of set (16) equivalently describe the macrostate of the system and that
(18) 
(19) 
may be considered nonequilibrium equations of state, there is a close analogy with the intensive thermodynamic variables in equilibrium.
After presenting the relevant variables and the nonequilibrium statistical operator, the next step in the thermodynamical description is the derivation of the evolution equations of the thermodynamical variables in set (7). Such equations form a system of nonlinear coupled integrodifferential equations which is discussed in Ref. vannucchi2010 () and in a detailed form in Ref. vannucchi2013 (). As stated there, for specific spin systems, it suffices to follow the evolution of magnons’ populations ; moreover for the equation of state it follows that
(20) 
or, alternatively,
(21) 
We recall that the equations of evolution for the populations are the quantum mechanical equations of motion for the dynamical quantities averaged over the nonequilibrium ensemble. They are handled resorting to the NESEFbased nonlinear quantum kinetic theory, with the calculations performed in the approximation that incorporates only terms quadratic in the interaction strength  with memory and vertex renormalization neglected, that is, we keep what in kinetic theory is called the irreducible part of the twoparticle collisions 
(22) 
(23) 
(24) 
(25) 
with and being the variables of sets (8) and (7) respectively, and
(26) 
stands for functional differentiation.
In a compact form we may write
(27) 
where
(28) 
is the source term that accounts for the pumping of energy to the system, stands for the population of photons of the source;
(29) 
is a nonlinear term of interaction between the spin subsystem and the blackbody radiation ( being its photon’s population);
(30) 
is the linear relaxation to the lattice with characteristic time . The last two terms are nonlinear contributions;
(31) 
the socalled Fröhlich term, a nonlinear interaction between magnons mediated by the lattice, and
(32) 
accounts for the magnonmagnon scattering interaction term.
Although the kinetic equations for the populations [Eq. (27)] may well describe the thermodynamic evolution of the magnetic subsystem, the complete thermodynamic description of the entire system must also include the evolution of the energy of the thermal bath (lattice and blackbody radiation). In a similar form of Eq. (22) we have that
(33) 
It is simple to show that and are null. The last term is composed by two contributions:
(34) 
the first,
(35) 
is the contribution which accounts for the thermal diffusion to the reservoir with a thermal diffusion time and tends to lead the thermal bath to equilibrium (characterized by the equilibrium energy ). The other contribution is related to the energy received from the subsystem of magnons.
Our system has its thermodynamical evolution described by the kinetic equations (27) and (33) and they must be solved. Since we stated before that the thermal bath is in a stationary state near the equilibrium condition defined by the external reservoir we have that
(36) 
and the thermal diffusion effect is sufficiently rapid for keeping this configuration. In this case , and .
Considering again the evolution of the population of magnons, we emphasise that Eq. (27) constitutes a nonlinear system of coupled integrodifferential equations. Its resolution in an approximate form called “two fluid model” is discussed on Refs. vannucchi2010 (); vannucchi2013 (), where the mean populations and were defined representing the populations of magnons around the minimum of frequency and those being fed by the external source respectively,
(37) 
and are the correspondent regions in the reciprocal space. Their evolution equations were obtained from Eq. (27),
(38a)  
(38b)  
(38c) 
and
(39a)  
(39b)  
(39c)  
(39d) 
where is the scaled time , taking the relaxation time as having a unique constant value (independent), are the populations in equilibrium, and and the fractions of the Brillouin zone corresponding to the two regions in the twofluid model. Moreover, the coefficients and are the coupling strengths associated to magnonmagnon interaction and to Fröhlich contribution respectively, is the one associated to decay with emission of photons, and is an average population of the phonons. Finally, the parameter is related to the rate of the rfradiation field transferred to the spin system, whose absorption is reinforced by a positive feedback effect. All these coefficients are dimensionless, being multiplied by the relaxation time .
In a similar fashion, the energy of the thermal bath has an evolution given, in the two fluid model, by
(40) 
being and the energy of the magnons in the regions and , and .
On Fig. 1 we show the evolution of the populations and , departing from equilibrium, under the action of the pumping source (we adopted for comparison with experimental data demokritov2006 ()), solving numerically Eqs. (38) and (39). As stated on Refs. vannucchi2010 (); vannucchi2013 (), besides the good agreement with the experimental data, this result shows clearly the accumulation of magnons on the mode of minimum frequency ().
Moreover, the analysis of the steady state of the system, i. e., the solutions of Eqs. (38) and (39) such that and are null, make evident the role of the Fröhlich term to the condensation of magnons. On Fig. 2 we show the values of the steadystate populations, and as a function of the scaled rate of pumping , and it can be noticed the existence of two pumping scaled rate thresholds, the first, after which there follows a steep increase in the population of the modes lowest in frequency, corresponds to the emergence of BEC, while the second, for higher values of , accounts for the internal thermalization of the magnons which acquire a common quasitemperature, implying that the magnonmagnon interaction overcomes Fröhlich contribution and BEC is impaired.
The stability of these solutions was analyzed firstly through the evaluation of the Lyapunov exponents. Defining the variables , , and in such manner that
(41)  
(42)  
(43)  
(44) 
we have that the Lyapunov exponents are the solutions of
(45) 
or
(46) 
whose solutions are
(47) 
shown in Fig. 3 as function of .
Besides the expected negative values, that reflects the stability of these solutions, it is interesting to note the two peaks occurring precisely for the values associated with the intensity thresholds, indicating, in these regions, a possible instability of the thermodynamic branch if we vary the values of the parameters of Eqs. (38) and (39).
Iii Informational Statistical Thermodynamics of NEFBEC
We proceed to the description of the NESEFbased Informational Irreversible Thermodynamics, ISTluzzi2000 (); luzzi2001 (), of the NEFBEC of magnons demokritov2006 (); demidov2008 () with the description given in Refs. vannucchi2010 (); vannucchi2013 (), and summarized in the previous section.
iii.1 IST entropy
Here, in the framework of IST, we introduce the informational entropy
(48) 
where, we recall, is the nonequilibrium statistical operator of Eq. (12) and is a timedependent projection operator (it is characterized by the nonequilibrium state of the system at any time ) such thatluzzilivro2002 (); luzzi1990 (); luzzi2000b (); balian1986 ()
(49) 
and
(50) 
where and are those of Eqs. (15) and (13) but in the contracted description that takes as relevant microvariables, as stated before, only the occupationnumber operator of magnons, and the Hamiltonian of the thermal bath,, that is, in Eq. (15) the terms involving , , , and their conjugates are neglected.
Hence we have that
(51) 
(52) 
where and are the canonical and nonequilibrium partition functions [see Eqs. (13) and (17)]. The last one depends on time and must be explicitly written in terms of the nonequilibrium thermodynamic variables. In a analogous way to the equilibrium Bosestatistics we obtain that
(53) 
Thus
(54) 
and using the relation between and , Eq. (21), one obtains the expression for the informational entropy
(55) 
whereas