# Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results

## Abstract

We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L{yields}{infinity} and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx{sup -(d+{sigma})} as x{yields}{infinity}, with 2<{sigma}<4 and 2<d+{sigma}{<=}6, on the Casimir effect at and near the bulk critical temperature T{sub c,{infinity}}, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged--i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force F{sub C}. We generalize the phenomenological scaling Ansaetze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form L{sup d}F{sub C}/k{sub B}T{approx_equal}{xi}{sub 0}(L/{xi}{sub {infinity}})+g{sub {omega}}L{sup -{omega}}{xi}{sub {omega}}(L/{xi}{sub {infinity}})+g{sub {sigma}}L{sup -{omega}{sub {sigma}}}{xi}{sub {sigma}}(L/{xi}{sub {infinity}}). Here {xi}{sub 0}, {xi}{sub {omega}}, and {xi}{sub {sigma}} are universal scaling functions; g{sub {omega}} and g{sub {sigma}} are scaling fields associated with the leading corrections to scaling and those of the long-range interaction,more »

- Authors:

- Fachbereich Physik, Universitaet Duisburg-Essen, Campus Essen, 45117 Essen (Germany)
- (Bulgaria)

- Publication Date:

- OSTI Identifier:
- 21069754

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; Journal Volume: 73; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevE.73.016131; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; CASIMIR EFFECT; CORRECTIONS; CORRELATIONS; CRITICAL TEMPERATURE; FREE ENERGY; GAUSSIAN PROCESSES; HAMILTONIANS; INTERACTION RANGE; PAIRING INTERACTIONS; PERIODICITY; PHASE TRANSFORMATIONS; POTENTIALS; RENORMALIZATION; SINGULARITY; SLABS; SPHERICAL MODEL; VAN DER WAALS FORCES

### Citation Formats

```
Dantchev, Daniel, Institute of Mechanics, Bulgarian Academy of Sciences, Academic Georgy Bonchev Street Building 4, 1113 Sofia, Diehl, H. W., and Grueneberg, Daniel.
```*Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVE.73.016131.

```
Dantchev, Daniel, Institute of Mechanics, Bulgarian Academy of Sciences, Academic Georgy Bonchev Street Building 4, 1113 Sofia, Diehl, H. W., & Grueneberg, Daniel.
```*Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results*. United States. doi:10.1103/PHYSREVE.73.016131.

```
Dantchev, Daniel, Institute of Mechanics, Bulgarian Academy of Sciences, Academic Georgy Bonchev Street Building 4, 1113 Sofia, Diehl, H. W., and Grueneberg, Daniel. Sun .
"Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results". United States. doi:10.1103/PHYSREVE.73.016131.
```

```
@article{osti_21069754,
```

title = {Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results},

author = {Dantchev, Daniel and Institute of Mechanics, Bulgarian Academy of Sciences, Academic Georgy Bonchev Street Building 4, 1113 Sofia and Diehl, H. W. and Grueneberg, Daniel},

abstractNote = {We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L{yields}{infinity} and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx{sup -(d+{sigma})} as x{yields}{infinity}, with 2<{sigma}<4 and 2<d+{sigma}{<=}6, on the Casimir effect at and near the bulk critical temperature T{sub c,{infinity}}, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged--i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force F{sub C}. We generalize the phenomenological scaling Ansaetze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form L{sup d}F{sub C}/k{sub B}T{approx_equal}{xi}{sub 0}(L/{xi}{sub {infinity}})+g{sub {omega}}L{sup -{omega}}{xi}{sub {omega}}(L/{xi}{sub {infinity}})+g{sub {sigma}}L{sup -{omega}{sub {sigma}}}{xi}{sub {sigma}}(L/{xi}{sub {infinity}}). Here {xi}{sub 0}, {xi}{sub {omega}}, and {xi}{sub {sigma}} are universal scaling functions; g{sub {omega}} and g{sub {sigma}} are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; {omega} and {omega}{sub {sigma}}={sigma}+{eta}-2 are the associated correction-to-scaling exponents, where {eta} denotes the standard bulk correlation exponent of the system without long-range interactions; {xi}{sub {infinity}} is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution {proportional_to}g{sub {sigma}} decays for T{ne}T{sub c,{infinity}} algebraically in L rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and L. We derive exact results for spherical and Gaussian models which confirm these findings. In the case d+{sigma}=6, which includes that of nonretarded van der Waals interactions in d=3 dimensions, the power laws of the corrections to scaling proportional to b of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy {omega}={omega}{sub {sigma}}=4-d that occurs for the spherical model when d+{sigma}=6, in conjunction with the b dependence of g{sub {omega}}.},

doi = {10.1103/PHYSREVE.73.016131},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

number = 1,

volume = 73,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}