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Statistical physics in deformed spaces with minimal length

Explore the implications of minimal length on statistical physics in deformed spaces. Study deformed algebras, coordinate uncertainty, and the effect on heat capacity. Example of harmonic oscillators. Conclusions on the decrease of heat capacity at high temperatures.

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Statistical physics in deformed spaces with minimal length

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  1. Statistical physics in deformed spaces with minimal length TarasFityo Department for Theoretical Physics, National University of Lviv

  2. Outline • Deformed algebras • The problem • Implications of minimal length • An example • Conclusions

  3. Coordinate uncertainty: Deformed algebras Kempf proposed to deform commutator: Maggiore: Maggiore M., A generalized uncertainty principle in quantum gravity, Phys.Lett. B. 304,65 (1993). Kempf A.Uncertainty relation in quantum mechanics with quantumgroupsymmetry, J. Math. Phys. 354483 (1994).

  4. The problem Statistical properties are determined by Classical approximation arecanonically conjugated variables.

  5. General form of deformed algebra It is always possible to find such canonical variables, that satisfy deformed Poisson brackets. Chang L. N.et al, Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem, Phys. Rev. D. 65, 125028 (2002).

  6. Jacobian Jcan always be expressed as a combination of Poisson brackets: D=1: D=2:

  7. Implications of minimal length If minimal length is present then or faster for large For large kinetic energy behaves as Schrödinger Hamiltonian: For high temperatures Kinetic energy does not contribute to the heat capacity. Minimal length “freezes” translation degrees of freedom completely.

  8. Example: harmonic oscillators One-particle Hamiltonian: Kemp’s deformed commutators: The partition function:

  9. Blueline – exact value of heat capacity Redline – approximate value of heat capacity Green line–exact value without deformation

  10. Blueline – exact value of heat capacity Redline – approximate value of heat capacity Green line–exact value without deformation

  11. Conclusions • We proposed convenient approximation for the partition function. • It was shown that minimal length decreased heat capacity in the limit of high temperatures significantly.

  12. Dziękuję za uwagę!Thanks for attention! T.V. Fityo, Statistical physics in deformed spaces with minimal length, Phys. Let. A 372, 5872 (2008).

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