Entropy of a continuous system

In summary: This approach is used in quantum field theory, where the concept of entropy density is extended to the more general concept of entropy functional. In summary, the entropy of a continuous system, such as the electromagnetic field, can be defined using the entropy density and can be calculated using statistical mechanics principles.
  • #1
dEdt
288
2
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.
 
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  • #3
Obviously you can't use something like the log of the phase space volume,
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.
 
  • #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.
 
  • #5
Andy Resnick said:
If you can assign a temperature (which you can for blackbody radiation), you can define the entropy:

http://128.113.2.9/dept/phys/courses/PHYS4420/BlackBodyThermo.pdf

Thanks for the paper, but I'm looking for a more statistical approach.

Jano L. said:
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.

Jorriss said:
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.

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.
 
  • #6
dEdt said:
Thanks for the paper, but I'm looking for a more statistical approach.

Google is your friend:

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

Section 5.3
 
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  • #7
dEdt said:
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.

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.
 

1. What is the definition of entropy in a continuous system?

The entropy of a continuous system refers to the measure of disorder or randomness in that system. It is a quantitative measure of the number of possible configurations or states that a system can have.

2. How is entropy related to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system always increases over time or remains constant in ideal cases. Entropy is a key concept in this law as it represents the direction of natural processes towards a state of maximum disorder.

3. What are the units of entropy in a continuous system?

The units of entropy in a continuous system are typically Joules per Kelvin (J/K). This indicates the change in energy per unit temperature that is required to increase the disorder or randomness of a system.

4. How does the entropy of a continuous system affect its stability?

In general, an increase in entropy leads to a decrease in stability of a system. This is because higher entropy means more disorder, which can lead to unpredictable or chaotic behavior. However, there are cases where an increase in entropy can actually increase the stability of a system.

5. Can the entropy of a continuous system ever decrease?

According to the second law of thermodynamics, the total entropy of a closed system cannot decrease over time. However, certain subsystems within a larger system can experience a decrease in entropy if there is an input of energy from an external source. This decrease in entropy is offset by an increase in entropy in the surrounding environment.

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