Strong Coupling and non-Markovian Effects in the Statistical Notion of Temperature
We investigate the emergence of temperature in the system-plus-reservoir paradigm starting from the fundamental microcanonical scenario at total fixed energy where, contrary to the canonical approach, is not a control parameter but a derived auxiliary concept. As shown by Schwinger for the regime of weak coupling between system and environment, emerges from the saddle-point analysis leading to the ensemble equivalence up to corrections in the squared root of the inverse number of particles that defines the thermodynamic limit. By extending these ideas for finite , while keeping , we provide a consistent generalization of temperature in strongly coupled systems and we illustrate its main features for the specific model of Quantum Brownian Motion where it leads to consistent microcanonical thermodynamics. Interestingly, while this is a monotonically increasing function of the total energy , its dependence with is a purely quantum effect drastically different for Markovian and non-Markovian regimes.
In the context of statistical physics there are two ways to explain how a system acquires a property associated with the thermodynamic notion of temperature libroPathria (); Libro_Greiner (). In the first approach, one considers the system as weakly coupled with a thermal bath that is initially in a canonical state at temperature . If we wait long enough, will equilibrate (in the sense of stationarity of macroscopic observables) and acquire itself a canonical distribution at the same . Here, therefore, the idea of temperature is pre-assumed from the beginning. In the second approach, one considers instead that the global system is in a microcanonical distribution at total energy (we agree with HÃ¤nggi1 (); Campisi (); Franzosi () that this is the conceptually foundational starting point to understand the meaning of temperature). Here and equilibrate due to the presence of a weak interaction term, and the temperature will emerge as a parameter that fixes the condition of equilibrium. The temperature is then a derived rather than a fundamental quantity.
As it has been shown when going beyond the assumption of weak interactions, for strongly coupled and deviations from the standard thermodynamics emerge entropy (); entropy2 (); strong3 (); strong4 (), as well as problems defining local temperature effective (); HÃ¤nggi2 (); Temp2 (); therm_equil (). Also, the equivalence between the microcanonical and the canonical approach does not hold, correlations between system and bath become important, and the system is non-extensive by nature Strong_termo (); Nature (); Petruccione (). In this context it is well known that when is strongly coupled to a thermal bath the long time steady state of the system, contrary to the weak coupling scenario, does not take the Boltzmann form, neither in the open-quantum system approach Fleming (); Chem (); 23 (), in the global closed thermal state scenario canonical (), nor in the pure state setup popescue1 (); popescue2 ().
In this paper we will provide a consistent definition of temperature in the system-plus-bath scenario with arbitrary coupling strength by starting with a global microcanonical state at energy and generalizing the saddle point analysis of ensemble equivalence pioneered by Schwinger Schwinger (); libroQSFT (). In section II we review the relevant aspects of the emergence of temperature in the weak coupling scenario, then in section III we generalize this idea to the finite coupling case, and as an experimental relevant application, we will present the main features of this definition of temperature in the solvable case of Quantum Brownian Motion. We give some general conclusions in section IV and an outlook overview in the last section V.
Ii The statistical emergence of temperature from a saddle-point condition
Let us first review the weak coupling case by considering two initially isolated systems and with Hamiltonians and . They have fixed energies , and each of them is described by a microcanonical state
where denotes the density of states, and the trace over each Hilbert subspace . is the central quantity connecting statistics and thermodynamics through the Boltzmann equation
where is the Boltzmann constant and the thermodynamic entropy of each subsystem.
When we bring and into weak thermal contact, allowing them to interchange energy through a small interaction term such that , in the long time limit, , the resulting global state is microcanonical
with total energy . Here acts in and . We can write the density of states, expanding the operator-valued Dirac delta function pie (), as
where is an integration variable with units of inverse energy. Defining , Eq. (4) takes the form
with an associated action
where the decomposition with and is possible because the interaction term is small enough to be neglected. Following an idea due to Schwinger Schwinger (), for a large number of degrees of freedom , the quantities and are large (allowing rapid oscillations in the exponential) and we can solve the integral in Eq. (5) using the Saddle-Point-Approximation (SPA). The saddle-point condition
admits an analytical continuation over the lower half of the complex plane where we find the sole saddle-point , with satisfying
By interpreting the real solution as the inverse temperature and as the canonical partition function, then is the mean internal energy of each subsystem, and the relation
gives a condition on how the total energy is distributed between systems and when they are brought into contact. This microscopic analysis is thus used as the definition of both thermal equilibrium and of the inverse temperature that fixes this condition. To make this microscopic construction complete and see how can indeed be interpreted as the thermodynamic temperature, let us use Eqns. (2) and (5) with the solution established by Eq. (8) to obtain
which for a large number of degrees of freedom and by introducing the Helmholtz free energy , immediately gives the well known thermodynamic relation libroQSFT ()
If there exist many solutions to Eq. (7) over the imaginary axes, say: with , we must choose the solution in Eq. (10) in accordance with the principle of maximum entropy, and neglecting exponentially small corrections .
In the approach of Schwinger, the focus of the SPA analysis was to provide a way to justify the ensemble equivalence as follows. Following a similar procedure as the one leading to Eq. (5), we write down the microcanonical density matrix as
where we have multiplied and divided by . The expectation value of an observable acting on the global system can then be written in the form
where . In the thermodynamic limit, provided varies slowly with respect to , the integral in Eq. (13) can also be solved by SPA, resulting in , where is the solution of Eq. (8). This is the meaning of equivalence of ensembles, according to Schwinger, in the weak coupling approach.
After this revision of the key aspects of the emergence of temperature in composite weakly interacting systems, we now proceed to extend these ideas to systems with non-negligible interaction Hamiltonian .
The key point that allow for this generalization is that the SPA analysis that naturally leads to the concept of temperature is not restricted to at all, and in fact its only requirement is consistency with the thermodynamic limit .
Iii Finite coupling regime
While the global equilibrium state in the case of finite interaction energy is
and the density of states is still given by Eq. (5), now can not be unambiguously decomposed in general in terms of the bare Hamiltonians and spheat (). However, our key observation is that as long as we can solve the integral in Eq. (5) by SPA, the resulting real solution for , which now depends not only on the total energy but also on the parameters of the interaction, characterizes the condition of thermal equilibrium between and , thus providing the statistical definition of temperature for systems with finite coupling. To support this claim we will now study the consistency and consequences of this definition in a solvable example.
As a specific microscopic model that allows for almost full analytical treatment and remains of high experimental relevance, we consider now a microcanonical modification of the widely used open-system approach to Quantum Brownian Motion (QBM)Ingold (). Here consists of a quantum harmonic oscillator linearly coupled to a bath of non-interacting harmonic oscillators. The total Hamiltonian reads
where and are momentum and position operators of the coupled harmonic oscillator with bare frequency and mass , and the momentum and position operators of the th bath oscillator with frequency and mass coupled with the central system through characteristic fixed for a given model of the bath.
The bath and interactions are characterized by the bare and coupled spectral densities that distribute the frequencies like
with cut-off Drude frequency and so called damping parameter , which is a function of the parameters characterizing the system-bath coupling strength. The parameter is a characteristic of the bath with units of such that .
Here the imaginary temperature partition function of , using the spectral density from Eq. (16a), reads
where is the zero point energy of the bath, and is the Riemann zeta function. In this way we arrive at
where the effective , related to the coupled harmonic oscillator, has an explicit form in terms of Gamma functions Weiss (),
with and being the roots of the polynomial expression in that appears in the denominator of Eq. (17) and which carry the dependence on and .
Interestingly, as shown in pachon (), for systems that have an interaction that involves only relative coordinates, like in Eq. (15), the classical partition function does not depend at all on the coupling
and therefore the temperature is independent of , regardless how strong the interaction is. This means that for the model in Eq. (15) the dependence of the temperature on the coupling strength is purely a quantum effect.
Eq. (22) establishes the equilibrium relation for the total energy between systems and , where the l.h.s is related with the energy of the bath and the r.h.s accounts for the energy of the coupled oscillator.
In the regime where the r.h.s of Eq. (22) gives just the energy of the bare central harmonic oscillator: . In this case the energy of the bath composed of a big number of harmonic oscillators dominates over the energy of the single harmonic system and the derived temperature is given mostly by the bath contribution. This is the common scenario in the weak-coupling canonical approach, where the central system acquires a temperature given by the constant temperature of the canonical thermal bath. Here we will show that an interaction term that couples linearly the system with each degree of freedom of the bath gives rise to an interaction energy which affects the resulting equilibrium temperature.
but are readily renormalized by to obtain a global zero ground state energy. The new relation (22) for renormalized energy finally reads
where we have made use of the Vieta relation Weiss ().
The solution of Eq. (III) for (fixing the energy equilibrium condition for our model) accordingly defines the inverse temperature in the finite coupling regime.
This solution , our main result, depends on the interaction , the total energy , but also the bath parameters and , and the frequency .
Coming back to the issue of SPA vs weak coupling expansions, we stress again that in a model where the total energy scales with the number of particles , in solving Eq. (5) by SPA we are neglecting terms of , which is justified in the thermodynamic limit for large number of particles libroQSFT (). Still, the SPA approximation does not depend on any perturbative expansion of the interaction parameter and thus our results are valid beyond the weak-coupling limit.
In Fig. 1 we show the numerical solution of Eq. (III) for given values of and . As it can be observed, is a monotonous function of the total energy , with a decreasing tendency that is more pronounced close to the renormalized ground state. From Eq. (III), and using the asymptotic expansion of Digamma functions, we obtain that for , , as expected. Fig. 1 shows also the growing behavior of with , which is again more evident in the regime of small .
Remarkably, in the regime for large the dependence of with can be modulated to the extend of inverting the increase observed in Fig. 1. The tunning parameter is the ratio , that also quantifies the degree of memory of the bath or non-Markovianity paz (). This surprising connection between the dependence of the temperature on the coupling and the time scale associated to memory effects in the environment is shown in Fig. 2, and it is further explored by the explicit dependence of on the Drude frequency in Fig. 3.
We can provide a physical interpretation of the peculiar dependence of on the Drude frequency for our results in the following way:
In the Markovian regime Fig. 2 suggest that the action from the bath on the central system mostly determines the energy equilibrium condition: the energy-flow follows the natural direction from bath to system to reach equilibrium. Moreover, it can be shown that the r.h.s of (III) grows with in this regime. That explains the growing behavior of with in this case. On the other hand, in the non-Markovian regime Fig. 2 suggest that is the action from the system on the bath which determines the equilibrium condition: in this case the energy-flow direction goes from system to bath. Also, it can be shown that the contribution of the r.h.s of (III) in this case follows a decreasing or increasing behavior as a function of depending on the particular range of values of within the non-Markovian regime. This is reflected in the peculiar behavior appearing in Fig. 2.
We want to emphasize that the results obtained here go beyond any finite-order expansion around the weak-coupling scenario. Fig. 4 shows the contrast between the solution for temperature obtained in the first order expansion for in Eq. (III) and that obtained from the full expression. The characteristic saturation behavior clearly indicates the breakdown of any finite-order approximation in powers of .
Having obtained the inverse temperature we can calculate thermodynamic potentials for finite coupling. We start with Eq. (10), from which the entropy of the global system in the limit can be calculated by
Recognizing that is a function of , we get
and since , we finally obtain the thermodynamic relation
Following Ref. entropy3 () we may also calculate the entropy for the coupled oscillator as
where is taken from Eq. (20) and evaluated at the solution given by the SPA condition. The term is the thermodynamic mean energy of the coupled oscillator given by the r.h.s of Eq. (III) and of course also evaluated at the solution .
Fig. 5 illustrates the behavior of as a function of the total energy for various values of the coupling . As can be observed, is a positive quantity that becomes zero for , in nice accordance with the third law of thermodynamics. The entropy is also a monotonically increasing function of and . This latter feature accounts for the decrease in purity of the reduced density matrix with increasing .
Finally, let us consider the finite coupling version of the ensemble equivalence. Whenever the solution of the integral in Eq. (13) is justified by SPA, the saddle-point condition will give a relation between the expectation value of any smooth operator calculated in the microcanonical ensemble and the one calculated in the canonical case, but for a temperature given by the solution in the finite coupling regime. The same considerations also hold for the reduced density matrix describing the subsystem and, in that case, the relation for the expectation value of an observable is given as
that provides the sought extension of the equivalence of ensembles for systems with finite coupling in the thermodynamic limit. Accordingly, in Fig. 6 we show the expectation value of the squared position operator of the coupled oscillator evaluated at the solution . As expected grows with the energy and the particle is getting more localized with the increase of the damping parameter , as the bath monitors the position of the central particle carlos ().
It is interesting to note that a similar behavior of the entropy and the expectation value of the squared position with respect to in the QBM model is found in the canonical thermal bath approach entropy3 (); Weiss (). This is actually a non-trivial result due to the fact that our microcanonical thermodynamics is based on , which is a SPA condition solution that involves . One could imagine a situation where an observable in Eq. (13) has a non-smooth dependence with the integration variable , and therefore this dependence must by included in the SPA condition. In such scenario the equilibrium temperature becomes itself a function of the particular observable and our notion of ”ensemble equivalence” must be accordingly modified.
Starting from the fundamental microcanonical distribution we have studied the emergence of temperature in the finite coupling regime of open quantum systems. Following the approach pioneered by Schwinger resulting in as an emergent quantity which establishes the condition of equilibrium between two weakly coupled subsystems at energy , we have shown that in the finite and strong coupling regime also depends on the parameter that characterize the strength of the interaction. We have applied this idea to the paradigmatic Quantum Brownian Motion model and study the main features of this notion of temperature, confirming that is a monotonically decreasing function of the total energy , while the entropy of the coupled oscillator, which now depends on , is a positive quantity that starts from zero for and increases monotonically with and . Remarkably, we found also an unexpected dependence on the memory properties of the bath: while decreases as a function of the interaction parameter in the Markovian regime, this behavior is inverted in the non-Markovian case.
In a future work we want to extend this formalism to the cases where the central system is composed of non-interacting bosons or fermions which, besides energy, can also interchange particles with a reservoir. When a microscopic model is identified one must first extend the present formalism to the many-body context including correct fermion/boson statistics. Then, one can obtain the change of the temperature with respect to the coupling parameter, in the spirit of Fig. 2, and see how much the value of the temperature deviates from the temperature of the bath alone, a deviation that can be measured. In this case our formalism could be connected to recent experiments where thermodynamic quantities of few particles has been measured experiment (). This, and the effect of finite coupling in the degeneracy of the ground state of the isolated system degeneracy1 (); degeneracy2 () remain interesting questions for further study.
Acknowledgments. C.A.M. acknowledges financial support from the German Academic Exchange Service DAAD. We want to thank Josef Rammensee for useful discussions, and an anonymous referee for interesting suggestions.
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