582-2008-1 - Physics 582, Fall Semester 2008 Professor...

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Unformatted text preview: Physics 582, Fall Semester 2008 Professor Eduardo Fradkin Problem Set No. 1: Classical Field Theory Due Date: September 12, 2008 1 The Landau Theory of Phase Transitions as a Classical Field Theory In the Landau-Ginzburg approach to the theory of phase transitions, the ther- modynamic properties of a one-component classical ferromagnet in thermal equi- librium are described by a free energy functional of an order-parameter field ( vectorx ) ( the local magnetization). This functional contains, in addition to gradient terms, contributions proprtional to various powers of the local order parame- ter. Under some circumstances the coefficient of the 4 term of the energy functional may become negative. This is what happens if the local magnetic moments have spin-1 rather the spin- 1 2 . In this case, we have to include, in the energy functional, a term with a higher power of ( such as 6 ) in order to insure the thermodynamic stability of the system. The (free) energy density E for this system has the form E = 1 2 parenleftBig vector ( vectorx ) parenrightBig 2 + U ( ( vectorx )) where the potential U ( ( vectorx )) is U ( ( vectorx )) = m 2 2 2 ( vectorx ) + 4 4! 4 ( vectorx ) + 6 6! 6 ( vectorx ) . with m 2 = a ( T T ) and 4 < 0, 6 > 0. 1. Use a variational principle to derive the saddle-point equations ( i.e., the Landau-Ginzburg equations) for this system. 2. Plot the potential U ( ) for a constant field for 4 < 0 ( and fixed) at several temperatures. Show that, as the temperature T is lowered, there exists a temperature T > T at which the state with lowest energy has ( ) negationslash = 0 (for fixed 4 < 0 and 6 > 0). Plot the qualitative behavior of0)....
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This note was uploaded on 09/20/2010 for the course PHYS 582 taught by Professor Leigh during the Fall '08 term at University of Illinois, Urbana Champaign.

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582-2008-1 - Physics 582, Fall Semester 2008 Professor...

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