Advertisement

Applied physics

, Volume 3, Issue 1, pp 21–29 | Cite as

The long-time behaviour of the electric-field autocorrelation function in a finite photon gas

  • H. P. Baltes
  • E. R. Hilf
  • M. Pabst
Contributed Papers

Abstract

We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC 0(t)=At−4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k BT≲L and at timest withct≲L, C E(t) is described by an asymptotic expansion in powers ofL −1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC E(t) in terms of exp(−L −1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.

Index Headings

Finite system Thermodynamic limit Coherence function 

List of symbols

t

time

Cx

autocorrelation function of the stochastic variablex

ϱ

mass density

v

particle velocity

m

electric dipole moment

E

electric field

Tr

trace of operator or second order tensor

P

statistical operator

d

dimension

CE

coherence function=electric autocorrelation function

ij

complex electric second order correlation tensor

r1,r2

position vectors

Êi

i-th component of the electric-field operator

k

wavevector,k=|k|

L

edge length of the cube-shaped cavity

ni

non-negative integer

TM

transverse magnetic

TE

transverse electric

H

magnetic field

âk+

creation operator of photon withk

âk

annihilation operator of photon withk

h

Planck's constant

c

velocity of the light

V

volume of the cavity

ħ

h/2π

α

hc/k BT

T

temperature

kB

Boltzmann's constant

ij

complex magnetic second order correlation tensor

Uel

electric field energy

Umag

magnetic field energy

UTM

electric and magnetic field energy due to TM waves

UTE

ditto due to TE waves

U(T)

total field energy

ω

ck=angular frequency

B(ω)

Bose-Einstein distribution

δ(ω)

Dirac's delta distribution

DTM(k)

mode density due to TM waves

DTE(k)

ditto TE waves

D0(k)

Planck's mode density

ϱ0(k,T)

Planck's spectral energy density

τ

tk BT/h=reduced time

C0

thermodynamic limit ofC E

ζ(s, z)

generalized Riemann zeta function

ℒ(x)

Langevin function

fs(τ)

real part ofζ(s, 1+iτ)

CH

magnetic autocorrelation function

NTM(k)

number of TM modes with wavevectors not exceedingk

NTE(k)

ditto for TE modes

Aq(x)

number of lattice points in aq-dimensional sphere of radiusx 1/2

Γ(x)

Euler's gamma function

Ø

order of asymptotic behaviour in the average

QE, QH

normalization factors ofC E, CH

ζ(s)

Riemann zeta function

Λ

hc/k BT=“thermal” length

θ1

h/k BT=“thermal” time

θ2

(hL/ck BT)1/2=characteristic time

U0(T)

total field energy in the thermodynamic limit (Stefan-Boltzmann)

n

(n 1, n2, n3)

|n|

(n 1 2 +n 2 2 +n 3 2 )1/2

t0

period ofC E in the limit asTL→0

τ0

t 0kBT/ħ=reduced period

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Zwanzig: Ann. Rev. Phys. Chem.16, 67 (1965)CrossRefGoogle Scholar
  2. 2.
    B. J. Berne, G. P. Harp: Adv. Phys. Chem.12, 67 (1970)Google Scholar
  3. 3.
    B. J. Alder, T. E. Wainwright: Phys. Rev. A1, 18 (1970)CrossRefADSGoogle Scholar
  4. 4.
    P. Mazur: Physica Norvegica5, 291 (1971)Google Scholar
  5. 5.
    J. L. Lebowitz, J. Sykes: J. Stat. Phys.6, 157 (1972)CrossRefADSGoogle Scholar
  6. 6.
    J. R. Dorfman, E. G. D. Cohen: Phys. Rev. Letters25, 1254 (1970)CrossRefADSGoogle Scholar
  7. 7.
    R. Zwanzig, M. Bixon: Phys. Rev. A2, 2005 (1970)CrossRefADSGoogle Scholar
  8. 8.
    M. H. Ernst, E. H. Hauge, J. M. J. van Leeuwen: Phys. Rev. Letters25, 1254 (1970)CrossRefADSGoogle Scholar
  9. 9.
    R. J. Glauber: Phys. Rev.130, 2529 (1963) and131, 2766 (1963)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    C. L. Mehta, E. Wolf: Phys. Rev.134, A 1149 (1964)MathSciNetADSGoogle Scholar
  11. 11.
    E. F. Keller: Phys. Rev.139, B 202 (1965)CrossRefADSGoogle Scholar
  12. 12.
    K. Pöschl:Mathematische Methoden der Hochfrequenztechnik (Springer Verlag, Berlin, Heidelberg, New York 1956)zbMATHGoogle Scholar
  13. 13.
    R. C. Bourret: Nuovo Cimento18, 347 (1960)CrossRefGoogle Scholar
  14. 14.
    Y. Kano, E. Wolf: Proc. Phys. Soc.80, 1273 (1962)CrossRefGoogle Scholar
  15. 15.
    C. L. Mehta, E. Wolf: Phys. Rev.134, A 1143 (1964)MathSciNetADSGoogle Scholar
  16. 16.
    R. Fürth: Proc. Roy. Soc. (Edinb.) A67, 289 (1967)Google Scholar
  17. 17.
    F. H. Brownell: J. Math. Mech.6, 119 (1957)MathSciNetzbMATHGoogle Scholar
  18. 18.
    K. M. Case, S. C. Chiu: Phys. Rev. A1, 1170 (1970)CrossRefADSGoogle Scholar
  19. 19.
    H. P. Baltes, F. K. Kneubühl: Helv. Phys. Acta45, 481 (1972) Dissertation No. 4776, ETH Zürich 1971)MathSciNetGoogle Scholar
  20. 20.
    H. P. Baltes: Appl. Phys.1, 39 (1973)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • H. P. Baltes
    • 1
  • E. R. Hilf
    • 2
  • M. Pabst
    • 1
  1. 1.Institut für Theoretische Physik II (Kondensierte Materie)Freie UniversitätBerlin 33
  2. 2.Institut für KernphysikTechnische HochschuleDarmstadtFed. Rep. Germany

Personalised recommendations