Download Statistical physics in deformed spaces with minimal length.

Statistical physics
in deformed spaces
with minimal length
Taras Fityo
Department for Theoretical Physics,
National University of Lviv
Outline
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Deformed algebras
The problem
Implications of minimal length
An example
Conclusions
Deformed algebras
Coordinate uncertainty:
X  X min 

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 quantum group symmetry, J. Math. Phys.
35 4483 (1994).
The problem
Statistical properties are determined by
 En 
Z   exp   
 T 
n
Classical approximation
 H ( p, x ) 
Z   dpdx exp  

T


x, p are canonically conjugated variables.
General form of deformed algebra
 X i , Pj   i fij ( X , P)
 X i , X j   i gij ( X , P)
 Pi , Pj   i hij ( X , P)
 X , P   f ( X , P)
 X , X   g ( X , P)
P , P   h ( X , P)
i
j
i
i
ij
j
j
ij
ij
It is always possible to find such canonical
x, p variables, that X i  X i ( x, p), Pi  Pi ( x, p)
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).
H ( P, X )  H ( p, x)
 H ( p, x) 
Z   dpdx exp  

T


dPdX
 H ( P, X ) 
Z 
exp  

J
T


Jacobian J can always be expressed as
a combination of Poisson brackets:
D=1: J   X , P
D=2: J   X1 , P1 X 2 , P2    X1 , P2  X 2 , P1
 X1 , X 2 P1 , P2 
Implications of minimal length
If minimal length is present then J
or faster for large P.
P
D
n

P
For large P kinetic energy behaves as
  1/ 2m, n  2
Schrödinger Hamiltonian:
For high temperatures
Z  const
Z  const   ln T
Kinetic energy does not contribute to the heat
capacity. Minimal length “freezes” translation
degrees of freedom completely.
Example: harmonic oscillators
P
m
2
One-particle Hamiltonian: H 

X
2m
2
Kemp’s deformed commutators:


P
,
P
 X i , Pj   i 1   P 2   ij   PP
,
i
j
i j

  0,
2

2

X min 
3   
The partition function:
3/ 2 
2
2
 P  4 P dP
 2 T 
Z 
exp  

2 

J
 m  0
 2mT 
J  1   P
 1       P 
2 2
2
C
T
Blue line – exact value of heat capacity      0.01
Red line – approximate value of heat capacity
Green line– exact value without deformation
C
T
Blue line – exact value of heat capacity      0.01
Red line – approximate value of heat capacity
Green line– exact value without deformation
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.
Dziękuję za uwagę!
Thanks for attention!
T.V. Fityo, Statistical physics in deformed spaces
with minimal length, Phys. Let. A 372, 5872 (2008).