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Phase transitions in small systems: Microcanonical vs. canonical ensembles

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  • Dunkel, Jörn
  • Hilbert, Stefan

Abstract

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.

Suggested Citation

  • Dunkel, Jörn & Hilbert, Stefan, 2006. "Phase transitions in small systems: Microcanonical vs. canonical ensembles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 390-406.
    • Handle: RePEc:eee:phsmap:v:370:y:2006:i:2:p:390-406
    DOI: 10.1016/j.physa.2006.05.018
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    References listed on IDEAS

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    1. Adib, Artur B. & Moreira, André A. & Andrade Jr, José S. & Almeida, Murilo P., 2003. "Tsallis thermostatistics for finite systems: a Hamiltonian approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 322(C), pages 276-284.
    2. Chomaz, Ph. & Gulminelli, F., 2002. "Generalized definitions of phase transitions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 330-335.
    3. Gross, D.H.E., 2002. "Non-extensive Hamiltonian systems follow Boltzmann's principle not Tsallis statistics—phase transitions, Second Law of Thermodynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 99-105.
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    Cited by:

    1. Matty, Michael & Lancaster, Lachlan & Griffin, William & Swendsen, Robert H., 2017. "Comparison of canonical and microcanonical definitions of entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 474-489.
    2. Swendsen, Robert H. & Wang, Jian-Sheng, 2016. "Negative temperatures and the definition of entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 24-34.

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