1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Entropy of a continuous system

  1. Feb 20, 2013 #1
    How could the entropy of a continuous system, like the electromagnetic field, be defined? Obviously you can't use something like the log of the phase space volume, but I can't think of anything that would work.
  2. jcsd
  3. Feb 20, 2013 #2

    Andy Resnick

    User Avatar
    Science Advisor
    Education Advisor

  4. Feb 20, 2013 #3

    Jano L.

    User Avatar
    Gold Member

    Why do you think so? I think you may be right, since the number of Fourier variables (harmonic oscillators) is infinite, which makes the energy infinite.
  5. Feb 20, 2013 #4
    What's wrong with phase space volume? You can still write the Hamiltonian for a continuous system - it would just be a field theory now.
  6. Feb 21, 2013 #5
    Thanks for the paper, but I'm looking for a more statistical approach.

    Well, the phase space of a field is infinite dimensional. I wouldn't even know how to define volume, and if I could I'd think the volume of basically any region would be infinite.
  7. Feb 21, 2013 #6

    Andy Resnick

    User Avatar
    Science Advisor
    Education Advisor

    Google is your friend:

    http://home.comcast.net/~szemengtan/StatisticalMechanics/QuantumStatisticalMechanics.pdf [Broken]

    Section 5.3
    Last edited by a moderator: May 6, 2017
  8. Feb 22, 2013 #7


    User Avatar

    Use the entropy density (entropy per unit volume). It will, of course, not be conserved except for reversible situations. In general, there will be a rate of entropy creation per unit volume due to irreversible processes. If s is entropy density, then [tex]\frac{\partial s}{\partial t}+\nabla \mathbf{J}_s=\frac{\partial s_c}{\partial t}[/tex] where s is entropy density, [itex]\mathbf{J}_s[/itex] is the entropy flux, and [itex]\partial s_c/\partial t[/itex] is the rate of creation of entropy density (always non-negative).

    For example, for a simple fluid, the fundamental law says [itex]dU=T dS-P dV+\mu dN[/itex] where U is internal energy, T is temperature, S is entropy, P pressure, V volume, [itex]\mu[/itex] chemical potential, and N the number of particles. So it follows that [itex]dS=(1/T)dU+(P/T)dV-(\mu/T)dN[/itex] and in terms of densities: [tex]\frac{\partial s}{\partial t}=\frac{1}{T}\frac{\partial u}{\partial t}-\frac{\mu}{T}\frac{\partial n}{\partial t}[/tex] where u is internal energy density and n is particle density. And so forth.

    In statistical mechanics terms, you are considering each infinitesimal volume element to be an open equilibrated system. To find the total entropy, integrate the entropy density over the total volume.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook