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

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

3. Coordinate uncertainty: Kempf proposed to deform commutator : Maggiore : Deformed algebras Maggiore M., A generalized uncertainty principle in quantum gravity, Phys. Lett. B. 304, 65 (1993). Kempf A. Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35 4483 (1994).

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

5. 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). General form of deformed algebra It is always possible to find such canonical variables, that satisfy deformed Poisson brackets.

6. Jacobian J can 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 largekinetic 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 The partition function: Kemp’s deformed commutators: One-particle Hamiltonian:

9. Blue line – exact value of heat capacity Green line– exact value without deformation Red line – approximate value of heat capacity

10. Blue line – exact value of heat capacity Green line– exact value without deformation Red line – approximate value of heat capacity

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).