Glenn H. Fredrickson
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567295
- eISBN:
- 9780191717956
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567295.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter examines the subject of polymers experiencing external potential fields that vary in space. The statistical mechanical problem is reduced to the solution of Fokker-Planck equations, and ...
More
This chapter examines the subject of polymers experiencing external potential fields that vary in space. The statistical mechanical problem is reduced to the solution of Fokker-Planck equations, and operators for polymer segment density and stress are introduced. Analytical approximation schemes for solving the Fokker-Planck equations and evaluating operators are presented. Effective numerical methods are also described, with an emphasis on spectral collocation techniques.Less
This chapter examines the subject of polymers experiencing external potential fields that vary in space. The statistical mechanical problem is reduced to the solution of Fokker-Planck equations, and operators for polymer segment density and stress are introduced. Analytical approximation schemes for solving the Fokker-Planck equations and evaluating operators are presented. Effective numerical methods are also described, with an emphasis on spectral collocation techniques.
Glenn H. Fredrickson
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567295
- eISBN:
- 9780191717956
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567295.003.0005
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter discusses the topic of self-consistent field theory, a mean-field approximation method for analyzing the statistical field models constructed in Chapter 4. The mean-field approximation ...
More
This chapter discusses the topic of self-consistent field theory, a mean-field approximation method for analyzing the statistical field models constructed in Chapter 4. The mean-field approximation is introduced as a saddle point approximation to the functional integrals comprising a field theory model. The analytic structure of various models is discussed, along with the classification of saddle points and location in the complex plane. Further analytical approximations to simplify the mean-field equations are described, including weak inhomogeneity and slow gradient expansions, ground state dominance, and strong stretching approximations. Numerical methods for obtaining accurate and efficient solutions of the self-consistent field equations are presented.Less
This chapter discusses the topic of self-consistent field theory, a mean-field approximation method for analyzing the statistical field models constructed in Chapter 4. The mean-field approximation is introduced as a saddle point approximation to the functional integrals comprising a field theory model. The analytic structure of various models is discussed, along with the classification of saddle points and location in the complex plane. Further analytical approximations to simplify the mean-field equations are described, including weak inhomogeneity and slow gradient expansions, ground state dominance, and strong stretching approximations. Numerical methods for obtaining accurate and efficient solutions of the self-consistent field equations are presented.
