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Register a new userViolation of the Zeroth Law of Thermodynamics in Systems with Negative Specific Heat
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Added by
Abelardo Ram\\'irez
Publication date
2008
fields
Physics
and research's language is
English
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We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear, that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.
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The phenomenon described by our title should surprise no one. What may be surprising though is how easy it is to produce a quantum system with this feature; moreover, that system is one that is often used for the purpose of showing how systems equilibrate. The violation can be variously manifested. In our detailed example, bringing a detuned 2-level system into contact with a monochromatic reservoir does not cause it to relax to the reservoir temperature; rather, the system acquires the reservoirs level-occupation-ratio.
Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schr{o}dinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.
We study dipolarly coupled three dimensional spin systems in both the microcanonical and the canonical ensembles by introducing appropriate numerical methods to determine the microcanonical temperature and by realizing a canonical model of heat bath. In the microcanonical ensemble, we show the existence of a branch of stable antiferromagnetic states in the low energy region. Other metastable ferromagnetic states exist in this region: by externally perturbing them, an effective negative specific heat is obtained. In the canonical ensemble, for low temperatures, the same metastable states are unstable and reach a new branch of more robust metastable states which is distinct from the stable one. Our statistical physics approach allows us to put some order in the complex structure of stable and metastable states of dipolar systems.
The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of `equal temperature (ET), or `in equilibrium (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result cite{NEqFE} for Gaussian quantum systems shows that an effective temperature $T^{*}$ can be defined at all times during a systems nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the systems $T^*$ is slightly higher than the bath temperature $T_{textsc{b}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature $T_{textsc{b}}$ will not have the same equilibrated temperature as the first object. Thus $ET eq EQ $ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that `in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that `temperature appearing in thermodynamic relations becomes universally defined and makes better physical sense.
The theoretical model proposed by Einstein to describe the phononic specific heat of solids as a function of temperature consists the very first application of the concept of energy quantization to describe the physical properties of a real system. Its central assumption lies in the consideration of a total energy distribution among N (in the thermodynamic limit $N rightarrow infty$) non-interacting oscillators vibrating at the same frequency ($omega$). Nowadays, it is well-known that most materials behave differently at the nanoscale, having thus some cases physical properties with potential technological applications. Here, a version of the Einsteins model composed of a finite number of particles/oscillators is proposed. The main findings obtained in the frame of the present work are: (i) a qualitative description of the specific heat in the limit of low-temperatures for systems with nano-metric dimensions; (ii) the observation that the corresponding chemical potential function for finite solids becomes null at finite temperatures as observed in the Bose-Einstein condensation and; (iii) emergence of a first-order like phase transition driven by varying $N$.
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