Thermodynamics of Reduced State of the Field

Recent years have seen the flourishing of research devoted to quantum effects on mesoscopic and macroscopic scales. In this context, in Entropy 2019, 21, 705, a formalism aiming at describing macroscopic quantum fields, dubbed Reduced State of the Field (RSF), was envisaged. While, in the original work, a proper notion of entropy for macroscopic fields, together with their dynamical equations, was derived, here, we expand thermodynamic analysis of the RSF, discussing the notion of heat, solving dynamical equations in various regimes of interest, and showing the thermodynamic implications of these solutions.


I. INTRODUCTION
In recent years, a considerable attention has been given to the study of quantum phenomena on mesoscopic scale, as many physical systems that are nowadays fundamental for physical applications fall into this regime [1,2]. The main characteristic of mesoscopic systems is that, while they are still large enough not to be considered purely quantum, they are neither small enough to ignore quantum effects.
Furthermore, while the behavior of macroscopic fields is well described by classical wave equations with coherent sources, incorporation of thermal and random sources into the field equations still represents an open problem [3]. As a matter of fact the most common description of such a situation relies on the introduction of phenomenological terms, for example terms describing the damping. This solution is not fully satisfactory from a theoretical point of view, as these extra terms do not give a correct thermodynamic description of such systems. On this basis, and on drawing from the fact that the ultimate description of any physical system should be given by quantum mechanics, the Reduced State of the Field (RSF) formalism was conceived [4].
Since a completely quantum picture is in general too complex and consequently not convenient to treat macroscopic fields, the RSF aims at describing macroscopic waves using a coarse-grained version of the quantum formalism. Such description allows one to retain the most important quantum features that would even emanate at macroscopic scale [4], while at the same time mitigating the complexity, that would have no effect beyond the microscopic realm. Interestingly, in the same spirit one can answer the question being sort of opposite to the former one, namely, which features of the quantum evolution can be classified as classical [5].
On the other hand, recent years have seen the flourishing of quantum thermodynamics [6], namely, the study of thermodynamic phenomena on the quantum scale. This interest has been fostered by progressive miniaturization of electronic and optical devices, at the level where quantum phenomena cannot be ignored [7]. We therefore observe a huge development of the field of quantum thermodynamics, where a wide range of topics is being covered, e.g. thermalization [8,9], quantum heat engines and refrigerators [10][11][12][13], and quantum batteries [14][15][16]. It is thus of interest to see how thermodynamics intersects with the description of mesoscopic and macroscopic fields since, especially on the mesoscopic scale, one typically does not have full control over the system, yet quantum effects need to be taken into account in order to describe the system appropriately [17].
In this paper we want to explore how thermodynamic phenomena, such as heat exchange, fit the RSF formalism. Moreover, we want to analyze the behavior of the entropy of RSF [4], as its definition differs from the one usually found in the classical or quantum realms. The paper is organized as follows. In Sec. II we briefly review the RSF formalism, pointing out its main features. In Sec. III, starting from the evolution equations of RSF, we consistently define the main thermodynamic quantities, such as internal energy, heat and work. Then in Sec. IV we solve the equations of motion in some simple but relevant situations, highlighting the thermodynamic meaning of the different terms present therein. Finally, in Sec. V we give our conclusions and some outlooks for future works.

II. THE RSF FORMALISM
This section mostly follows [4], since we summarize here the most important background and ingredients of the RSF formalism. In particular, all formulas appearing in this section are taken from [4].
We start with classical electromagnetic field which in a finite volume is described by a set of modes f k (x; t) = e −iω k t f k (x), where x is the position, k is a discrete index and ω k is the frequency at which the mode oscillates. In the first quantization picture these modes represent eigenstates of the single-particle Hamiltonian of quasi-particles associated with the field. Under a proper normalization, these modes form an orthonormal basis of the single-particle Hilbert space, where the energy of each mode is equal to ω k .
In the second quantization picture, a pair of operatorsâ k ,â † k is associated to each mode f k . Standard bosonic commutation relations hold: so that the action of the annihilation and creation operators, on the vectors in the corresponding Fock space spanned by the orthonormal set {|n k }, iŝ The RSF formalism relies on a correspondence between operators acting on the single-particle Hilbert space and additive operators acting on the Fock space. The former can be written as: where |k ≡ |f k , while the corresponding additive observable in the Fock space is[? ] Consequently, unitary operatorsû acting on the single-particle Hilbert space are in correspondence with multiplicative operators on the Fock space via:û From now on we also use "Tr" for trace operations in the Fock space and "tr" for traces applied to the level of the RSF, i.e. on a single-particle Hilbert space. The RSF description of the state of a macroscopic field is based on the couple (r, |α ), defined from the full quantum state of the fieldρ in the Fock space as: The matrixr is a single-particle density operator, while the vector |α contains the information about the phase of the macroscopic field. It is important to observe that the single-particle density operator is not normalized to unity, but rather to the total number of particles in the state, i.e.: In fact, the same expectation-value identification holds for any additive observable tr rb = Tr ρB .
Furthermore, it turned out beneficial to define an another object, the correlation matrix which is a positive semi-definite operator being zero if and only if the state is coherent. Using this operator it is then possible to give a suitable definition of entropy for macroscopic fields, which is This definition of entropy has an appealing feature of being always greater than or equal to zero, being zero only when the RSF is coherent. This also highlights the fact that the coherent states are the only pure states in this formalism.
To shortly summarize the above, the RSF formalism is particularly suited to deal with situation where one does not have full quantum control of the system (we just control first and second moments, so to speak), as is in the case of macroscopic fields, but still quantum effects are visible. Having revised the RSF formalism and its main features, we are now ready to start thermodynamic considerations.

III. THERMODYNAMICS OF THE RSF
In a usual scenario described by thermodynamics one deals with a system S, often called the working fluid, interacting with one or more thermal baths, i.e. much larger systems with infinite heat capacity that are typically assumed to have a well defined temperature. By changing the Hamiltonian, i.e. the energy, of the working fluid S and letting it interact appropriately with the thermal baths, it is possible to extract work from the system (i.e. we have a heat engine) or to use work to transfer heat from a cold to a hot bath (i.e. we implement a refrigerator).
As in what follows we will not be interested in a description of the thermal baths, but rather in their action on the working fluid SS Therefore we want to define heat and work only in terms of the state S, in the current context sufficiently well described by the couple (r, |α ). In order to study the thermodynamics of a macroscopic field described under the RSF formalism, we first need to recall the dynamical equations describing the behaviour of the field when it interacts with an external bath. This was already done in [4], where the system of equation for the RSF was derived from the standard expression for a map belonging to a so called quasi-free dynamical semigroup [18,19], thus extending this concept to RSF formalism. The set of equations [4] describing the dynamics of the couple (r, |α ) can be written as[? ]: Let us start by explaining the meaning of each term in Eqs. (11a, 11b). In the dynamical equation forr we first find the commutator ofr with the single-particle Hamiltonianĥ = k ω k |k k|, and this term describes nothing but the standard unitary dynamics induced by the free Hamiltonian. Next we find the term |ζ α|, which describes the effect of a coherent source, and thus depends also on the phase of the system |α . Then we can see the anticommutator term with the operatorsγ , describing stimulated absorption and emission processes, while the isolated termγ ↑ describes spontaneous emission processes. The operatorsγ can be expressed as: where the coefficients Γ kk encode the information about the state of the thermal bath and its interaction with the system. Finally, the integral term describes the effect of random scattering phenomena, where the operatorsû are unitary. Similar considerations apply to the dynamical equation for |α .
As the entropy is defined in terms of the correlation matrixr (α) , it is also useful to derive the dynamical equation for this quantity. Sincer (α) =r − |α α|, we only need to compute the time derivative of |α α| using Eq. (11b): from which we can write the dynamical evolution for the correlation matrixr (α) as d dtr From this equation we can see that the dynamics of the correlation matrix is not influenced by the presence of coherent sources. Consequently, the entropy S[r (α) ] is also invariant with respect to coherent evolution. This feature of the theory is associated with the fact that we are dealing with a mesoscopic or macroscopic system, where in fact we do not have access to all degrees of freedom [4]. In particular, the single-particle Hamiltonianĥ does not carry the whole content of the Hamiltonian in the Fock space which also contains contributions due to the displacement. In view of this, we find it reasonable that the contribution to the heat stemming from the coherent source be zero as well, and we thus define the internal energy as Using this notion of internal energy, one has a natural decomposition Two observations are in place here. First of all, the single particle Hamiltonian is time independent by construction. This is because the frequencies as well as the eigenmode basis of the Hamiltonian are not under control and do not vary over time due to the dynamics of the sole field. Therefore, for generic macroscopic fields, there is no work. Just the heat. Work would require an engineered variant of time evolution, i.e. one can perform (extract) work on (from) the system only by changing the frequencies ω k .
Second of all, only the scattering term couplesr (α) with |α in Eq. 14. This feature in a salient way distinguishes the scattering processes from the other processes subsumed in the dynamical equations. Within a thermodynamic description, which is solely based here on the correlation matrix, the scattering belongs to a different (more complex) class of (likely non-equilibrium) processes. The latter property, however, would strongly depend on the measure µ(du) chosen. Perhaps, for the invariant Haar measure the situation would simplify, still, the aforementioned coupling will be there.
Therefore, we believe that the scattering processes deserve a separate and detailed treatment. Consequently, here we shall neglect random scattering terms, with the goal of delineating the heat exchange and entropy production due to other processes. Under this simplifying assumption the heat exchanged is equal to that is, it only depends on interactions with the thermal bath. The variation of the entropy in time is also found to be We use the notation in which the fraction of non-negative operators needs to be understood in terms of their eigenvalues. This is possible because whenever some eigenvalue approaches 0, the time derivative also vanishes, killing the potential singularities [20]. Note that the trace ofr (α) does not need to be constant in time. For a quasi-static process, in which the stateρ is always in thermal equilibrium, the correlation matrix is always of the form Since in this case we recover the equality from standard thermodynamics This observation further strengthens our definition of work and heat. Moreover, for a non quasi-static process one has thatr (α) is not of the form in Eq. (20), and thus one has also entropy production.

IV. SOME EXAMPLES OF RSF THERMODYNAMICS
In the following subsections we want to solve the dynamical equations (11a, 11b) under various circumstances where some of the terms are absent or can be simplified, thus highlighting their thermodynamic meaning.

A. Free dynamics of the RSF
The simplest, and almost trivial, case that one can analyze is the one where no interaction with either a coherent source or a thermal bath is present, so that the dynamics of the RSF is fully described by the Hamiltonian term alone. Assuming the Hamiltonian in the Fock space to be: we can explicitly write down the equations governing the matrix elements r kk (t) and the vector components α k (t) as: The solutions to these equations are easily found: These solutions imply that, under purely free dynamics, the populations stay constant, while the coherences among them rotate at a frequency equal to the detuning between the modes. Finally, the components of the phase vector |α rotate at the corresponding frequency. In accordance with Eq. (17) there is no heat exchange, as there is no thermal bath. An important fact to be noted is that, as in Eq. (14) the correlation matrix depends on the Hamiltonianĥ only through the commutator term, the entropy is unchanged under purely Hamiltonian dynamics, since the eigenvalues ofr (α) are left unchanged.

B. RSF dynamics in presence of a coherent source
We now want to solve Eqs. (11a, 11b) subject to a coherent source, but still without a thermal bath, so that we get: where |ζ = k ζ k |k . We can easily get the dynamical equations for the matrix elements: Solving the second equation first we get so that ther matrix elements are After we perform the integral we get Let us consider the case where the initial phase vector |α is null, i.e. α k (0) = 0 for all k. In this case the solution for the phase and the matrix elements ofr reads: The latter result, for the diagonal elements r kk (t), reduces to This implies that the populations oscillate around the average values r kk (0) + |ζ k | 2 /ω 2 k . Of course the correlation matrix remains constant (also if initial |α is not null), so does the entropy.

C. Dynamics of the RSF in presence of a coherent source and a thermal bath
Let us now consider the case where also a dissipation term is present, i.e. we want to analyze the case where the system interacts with both a coherent source and a heat bath. In this case the dynamical equations forr and |α are: where the operatorsγ have already been defined aŝ Let us remind that the matrix elements Γ kk are the particle creation and decay rates that can be derived using the Fermi golden rule. Under the typical Born, Markov and secular approximations, the operatorsγ become diagonal where the rates Γ k , due to the thermal character of the bath, are related via k B being the Boltzmann constant and T being the temperature of the heat bath. In this case the dynamical equations for the RSF become: where we have defined Z k = 1 − e −β ω k −1 . These equations are of the same form as Eqs. ( 27, 28). This can be noted by defining complex frequenciesω k = ω k − iΓ k ↓ /2Z k . In this notation we get:  so that one can immediately write down the solution to the second equation as:

= " A w R W T U B x B M K S E H n s u j 1 V x R H C z 4 I = " > A A A B / H i c b V C 7 T s N A E D y H V w i v A C X N i Q i J K t g B A W U E D W W Q y E M k V r S + b M I p 5 7 N 1 t 0 a K r P A V t F D R I V r + h Y J / w Q k p I D D V a G Z X O z t B r K
which implies that the phases α k are driven towards their steady-state values As for the matrix elements r kk one finds: and consequently It is of particular interest to see the steady values of the matrix elements r kk r steady From this steady-state solution, together with Eq. (45), we can compute the associated correlation matrixr (α) , for which one simply obtains: From this result one can see clearly what was already noted in [4], namely that in presence of random scattering (which is absent in this case) or a thermal environment with temperature different from zero, it is impossible to obtain a coherent state, and that only an initial pure state remains pure when the above conditions are met. Next we express the entropy of the steady state as a function of β (we set k B = 1) S[r (α)steady ](β) = tr (r (α)steady + 1) ln r (α)steady + 1 −r (α)steady lnr (α)steady = tr βĥr (α)steady + tr ln r (α)steady + 1 = βU + tr ln r (α)steady + 1 , as it can be found using Eq. (48) and going through some algebra. One can immediately see that the entropy depends on the temperature, both through the partition functions and the occupation numbers of the modes. We plot in Fig. 1 the entropy as a function of the temperature β for different values of the frequency. From the plot it can be seen that lower frequency modes have a greater entropy than the modes with higher frequency. The equation (49) can also be rearranged as: so that in this way we are driven to define the equilibrium free energy F eq F eq = − 1 β tr ln r (α)steady This is exactly the sum of the equilibrium free energies of each mode. We can then define the free energy as: F = U − β −1 S = tr r (α)ĥ − 1 β tr (r (α + 1) ln r (α + 1 −r (α) lnr (α) = tr r (α) ĥ − 1 β ln r (α) + 1 r α − 1 β tr ln r (α) + 1 (52) where we have introduced the non equilibrium free energy We thus see how, in the presence of a thermal bath, and using the definition of internal energy of Eq. (15), we are able to define in a reasonable way the free energy, both "in and out" of equilibrium.

V. CONCLUSIONS
In this paper we have explored how to define thermodynamic quantities in the RSF formalism, given its definition of entropy. We have also seen some examples of dynamical regimes that allowed us to explicitly compute the quantities of our interest such as energy, heat, work and other thermodynamic functionals.
Starting from the definition of entropy given in [4], we gave a reasonable definition of internal energy, heat and work. We have been able to show that in a quasi-static equilibrium process our definition of heat gives the proper increase of entropy, and then we have also defined the equilibrium and non-equilibrium free energy.
It would be interesting in the future to further explore how to describe other thermodynamic phenomena under this formalism, such as work extraction from heat engines and work storage in batteries. This would surely help to further clarify how thermodynamics should be described at mesoscopic scales, and to individuate possible issues to be solved in this regime. Last but not least, scattering terms deserve a careful, separate consideration.