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A thermodynamic inequality (from minimum work)

  1. Apr 4, 2012 #1
    EDIT: Turns out, the solution to my question is related to the determinant of a positive definite quadratic form.

    This is more or less straight from Landau's Statistical Physics Part 1 (3rd edition), Chapter 21.

    I don't understand how the inequality/condition (the last equation in this post) arises. I think it's a mathematical problem, but I wonder if I have ignored some physical reasoning.

    The minimum work, which must be done to bring a body in an external medium from its equilibrium state to any neighbouring state:
    [tex]
    \delta E - T_0 \delta S + P_0 \delta V > 0
    [/tex]
    where [itex]T_0[/itex] and [itex]P_0[/itex] are (constant) temperature and pressure of the external medium, and where for the temperature and the pressure of the body in equilibrium with the external medium we would have: [itex]T=T_0[/itex], and [itex]P=P_0[/itex]

    Regarding the energy as [itex]E=E(S,V)[/itex], and expanding [itex]\delta E[/itex] as a series:
    [tex]
    \delta E = \frac{\partial E}{\partial S}\delta S + \frac{\partial E}{\partial V}\delta V + \frac{1}{2} \left[ \frac{\partial^2 E}{\partial S^2} (\delta S)^2+ 2 \frac{\partial^2 E}{\partial S\partial V} \delta S\delta V + \frac{\partial^2 E}{\partial V^2} (\delta V)^2 \right]
    [/tex]

    Note that [itex]\frac{\partial E}{\partial S}=T[/itex] and [itex]\frac{\partial E}{\partial V}=P[/itex]. Inserting this into the first equation gives the inequality:
    [tex]
    \frac{\partial^2 E}{\partial S^2} (\delta S)^2 + 2 \frac{\partial^2 E}{\partial S\partial V} \delta S\delta V+ \frac{\partial^2 E}{\partial V^2} (\delta V)^2> 0
    [/tex]

    If we want this inequality to be true for any values of [itex]\delta S[/itex] and [itex]\delta V[/itex], the following conditions have to hold:
    [tex]
    \frac{\partial^2 E}{\partial S^2} > 0
    [/tex]

    [tex]
    \frac{\partial^2 E}{\partial S^2}\frac{\partial^2 E}{\partial V^2} - \left( \frac{\partial^2 E}{\partial S\partial V} \right)^2 > 0
    [/tex]

    Where does the last condition come from?
     
    Last edited: Apr 4, 2012
  2. jcsd
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