Slides - Piasecki, Jarosław

8/12/2019 Slides - Piasecki, Jarosaw

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CASIMIR FORCES INDUCED BYBOSE-EINSTEIN CONDENSATION

Jarosaw Piasecki

[email protected]

Institute of Theoretical Physics

Faculty of Physics

University of Warsaw

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MOTTO

Tis much better to do a little with certainty

& leave the rest for others that come after,

than to explain all things by conjecture

without making sure of any thing.

Isaak Newton

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HENDRIK BRUGT GERHARD

CASIMIR (1909 - 2000)

H. B. G. Casimir

"On the attraction between two perfectly conductingplates"Proc. K. Ned. Akad. Wet. 51, 793-795 (1948).

cr4

2

240

H. B. G. Casimir, D. Polder

"The influence of retardation on the London-van derWaals forces"Phys. Rev. 73, 360-372 (1948).

U(r) = 12

r723c

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TYPES OF THE CASIMIR EFFECT

electromagnetic

in quantum field theory

in particle physics

in cosmologyin critical phenomena

dynamical Casimir effect

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STATING THE PROBLEM:

Derive the Casimir effect in an imperfect (interacting) Bose

gas filling the volume contained between two infiniteparallel plane walls.Hamiltonian of the imperfect Bose gas:

H=H0+ aV

N2

2

H0=kinetic energy (perfect gas Hamiltonian)a/V >0=repulsive mean-field interaction per pair ofbosonsV =volume occupied by the system.H is superstable!

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METHOD OF ANALYSIS

Bose gas occupies volumeV =L2Dof a rectangular box

with linear dimensionsL LD.Ddenotes the distance between twoL Lsquare walls.The excess grand canonical free energy per unit wall areais defined by

s(T , D , ) = limL

(T , L , D , )

L2

D b(T, )

whereb(T, )denotes the grand canonical potential perunit volume evaluated in the thermodynamic limit.The Casimir force equals

F(T , D , ) = s(T , D , )D

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BOUNDARY CONDITIONS

One-particle kinetic energy(k) = (k2x+k2y+k

2z)

2/2m

z-axis perpendicular toL Lwallsperiodic

kz =2

D

nz, nz = 0,

1,

2,...

Dirichlet

kz =

Dnz, nz = 1, 2,...

Neumann

kz =

Dnz, nz = 0, 1, 2,...

kx, ky -periodic b.c.

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IMPORTANT RELATION

Grand canonical potential

(T , L , D , ) = kBTln (T , L , D , )(T , L , D , )is related to the analytic continuation of theperfect gas partition function0 by

(T , L , D , ) = exp

L2D

2a 2

L2D

2a

(i) +ii

dt exp

L2D

a

t2

2 t

0(T , L , D , t)

(


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BULK PROPERTIES OF THE

IMPERFECT BOSE GAS

The bulk grand canonical free-energy density

b(T, ) = limL

1L3

kBT ln(T,L,L,) = p(T, )

can be calculated with the use of the steepest descent

method.If < c=an0,c

p(T, ) =

1

2 an2

(T, ) +p0(T, an(T, ))wheren(T, )is the unique solution of the equation

n=n0(T, an)KMMF, 08.11. 2012 p. 9/21


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BULK PROPERTIES OF THE

IMPERFECT BOSE GAS

If > c=an0,c

p(T, ) =2

2a+p0(T, 0)

In the two-phase region

n=

a

and the density of condensate is equal to

a n0,c

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CASIMIR FORCE: PERIODIC

BOUNDARY CONDITIONS

The steepest descent method yields the asymptotic form ofthe excess free energy density. The Casimir force in theone-phase region near the condensation point equals

F(T , D , )

kBT

=

1

D3

[2(x)

x (x) ]

with

(x) =

n=1

1 + 2nx

n3 exp(

2nx)

x= D

per, per = =

anc(anc

)

21/2

(3/2)

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CASIMIR FORCE IN THE PRESENCE

OF CONDENSATE

In the two-phase region (in the presence of condensate)one observes a power-law decay

F(T , D , )kBT =2(3) 1D3 , > anc

exactly the same, and with the same amplitude as in the

perfect Bose gas.

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IMPERFECT AND PERFECT GAS:

COMPARING CRITICAL BEHAVIOR

Divergence of the range of exponential forces at theapproach to condensation:

imperfect (mean-field) Bose gas

(anc )1

perfect Bose gas

0 ()1/2

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ONE PARTICLE DENSITY MATRIX


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ONE-PARTICLE DENSITY MATRIX

FOR=

/KBT >0

THE CASE OF A PERFECT GAS:

< x2|1|x1 >=F(|x2 x1|)

3F(x) =

j=1

1

j3/2

exp j x2

j2

=

xexp2

x

+

s=1

xexp A

+(s)x

2cos A(s)

x

with

A(s) =

2(2 + 42s2)1/4 1

(2 + 42s2)1/2 .

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BULK CORRELATION LENGTH AND

RANGE OF CASIMIR FORCES

Correlation function of a perfect Bose gas

6[n2(r; , T) n2] = j=1

1

j3/2exp

j r

2

j2

2

= /kBT, =h/

2mkBT

Large distance (r ) asymptotics

6[n2(r; , T) n2] =

r

2

exp

r

0()

.

0() = h4

2m1 = range of Casimir force !

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ELEMENTS OF CALCULATION

Knowledge of the asymptotic behavior of Bose functions

gr() =q=1

exp(q)qr

when 0

g1/2() =

, g1/2() = 1

permits to evaluate derivatives of the density with respectto the chemical potential at the condensation point=imp,c=nca.

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BULK CORRELATIONS AND


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BULK CORRELATIONS AND

CASIMIR FORCES

=range of Casimir forces.=range of bulk correlations

perfect gas

0,periodic= 20,Dirichlet = 20,Neumann=0

critical exponent= 1/2

imperfect (mean field) gasperiodic = 2Dirichlet = 2Neumann= 2

critical exponent= 1

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SOME REFERENCES

M. Krech, Casimir Effect in Critical Systems, World Scientific,

Singapour (1994).G. Brankov, N.S. Tonchev, and D. M. Danchev, Theory ofCritical Phenomena in Finite-Size Systems, World Scientific,Singapore (2000).

L. Palova, P. Chandra, P. Coleman, The Casimir effect fromcondensed matter perspective, Am. J. Phys. 77(2009) 1055.

E. Elizalde, A, Romeo, Essentials of the Casimir effect and itscomputation, Am. J. Phys. 59(1991) 711.

Ph.A. Martin, P.R. Buenzli, The Casimir effect,Acta.Phys.Polon.B 37(2006) 2503-2559.

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SOME REFERENCES

Ph. A. Martin and V. A. Zagrebnov, The Casimir effect

for the Bose-gas in slabs, EPL 73, 15 (2006).

M. Napirkowski and J. Piasecki , Phys. Rev. E 84,061105 (2011).

M. Napirkowski and J. Piasecki , J. Stat. Phys. 147,1145 (2012).

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