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Thermodynamics, a model of the symmetry restoration in the universe

  1. Jan 22, 2013 #1
    1. The problem statement, all variables and given/known data

    The energy density (u=E/V) of a thermodynamic system (used as a model of symmetry restoration in the early universe) is given by:

    [tex] u(T)=aT^4 + \Lambda(T)[/tex] where [tex] \Lambda =0, T \leq T_0 [/tex] or [tex] \Lambda =\Lambda_0, T>T_0 [/tex]

    [tex]k_B T_0 = 10^{14} GeV [/tex]

    a) Calculate the Helmholtz free energy for the system

    b) Calculate the pressure and entropy from a). To fix any constants of integration, use the condition [tex]p=aT^4/3 [/tex] at *very low *temperatures

    c) find the factor by which the volume changes if a container of this stiff is adiabatically and reversibly cooled from T just above T_0 to T just below T_0

    d) suppose that the system cools reversible to zero in a metastable phase in which [tex]\Lambda[/tex] remains stuck at [tex]\Lambda_0[/tex] (instead of going to zero below T_0). What are the values of energy and entropy in this limit?

    e) the system then spontaneously and rapidly decays to the stable state in which [tex]\Lambda=0[/tex]. Find the final temperature of the system and the entropy change of the transition.

    2. Relevant equations

    will be used in the next section


    3. The attempt at a solution

    a) The equation [tex]\left (\frac{\partial (F/T)}{\partial T}\right )_V=-\frac{E}{T^2}[/tex] should be used. After the integration, F/T is defined up to a constant F_0.

    b) F(T,V) from a) should be differentiated with respect to T and V.

    c) The equation [tex]dE(T,V)+p(T)dV=0[/tex] for adiabatic process should be used. It will give us the connection between T and V in this process.

    We know p(T) (from b)) and E(T,V).

    d)no idea

    e)no idea

    Am I doing something wrong or not?
     
    Last edited: Jan 22, 2013
  2. jcsd
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