A mathematically rigorous treatment of adiabatic processes?

In summary, the entropy of a quantum system does not change provided lambda is changed slowly enough between two values. This can be proven rigorously using the adiabatic theorem.
  • #1
petergreat
267
4
If a quantum system is subjected to a time dependent Hamiltonian with one parameter lambda, then its entropy does not change provided lambda is changed slowly enough between two values. How can this be proved rigorously? The http://en.wikipedia.org/wiki/Adiabatic_theorem" [Broken] is not enough, since this system is different in that it constantly re-thermalizes itself to a Boltzmann distribution in the process.

I know this is a standard topic in any statistical physics course, but I haven't seen one which is mathematically fully convincing to me.
 
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  • #2
Well let's give it a try,i don't know a damn about adiabatic theorem so let's try not to think about any theorem already quoted,but get to standard thermodynamic adiabatic process there your "parameter lambda",i expect it to show up as somehow related to "gamma, the ratio of specific heats" We know that the system has ZERO change in external interaction energy(Q) thus giving us some lead to follow, you must be knowing the thermodynamic treatment there, thenyou have to work your way backwards..
im most gratified if i am of any help here.
 
  • #3
If the system is finite, than any time dependence will be a unitary transformation, whether adiabatic or not, so that S is constant. If the system is infinite, than its spectrum is continuous and it will be very hard to establish an equivalent of adiabaticity.
 
  • #4
DrDu said:
If the system is finite, than any time dependence will be a unitary transformation, whether adiabatic or not, so that S is constant.

I'm not sure If I understand you. Even if the system is finite (composed of finitely many particles in a finite volume), an abrupt change (such as compression by a pistol) will disturb it and causes S to increase.
 
  • #5
A perturbation just means that the hamiltonian changes with time. Nevertheless, even a time dependent hamiltonian will transform a pure state into another pure state, at least for a finite isolated system. Hence there is no increase of entropy. That's Liouvilles theorem.
 
  • #6
DrDu said:
A perturbation just means that the hamiltonian changes with time. Nevertheless, even a time dependent hamiltonian will transform a pure state into another pure state, at least for a finite isolated system. Hence there is no increase of entropy. That's Liouvilles theorem.

Sure, every state goes to the corresponding state in the final configuration, if the Hamiltonian governs everything in our system, so the Gibbs entropy doesn't change. However, this is a thermal system, which constantly redistributes itself to a Boltzmann distribution. For example, if the initial energy levels have equal spacings, but the final energy levels are not equally spaced, then the initial occupation number for each energy level no longer satisfies the Boltzmann distribution for final energy levels. How can you show the entropy is constant even after this re-distribution?
 
  • #7
Yes, but then you should somehow build in this decoherence into your rigorous treatment.
 
  • #8
can we not correlate with thermodynamic model..would be simpler to understand although a bit more tiring to start off...??
 

1. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which there is no exchange of heat between a system and its surroundings. This means that the energy of the system remains constant, and any change in temperature or pressure is solely due to work done on the system.

2. How is an adiabatic process different from an isothermal process?

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. In contrast, an adiabatic process does not necessarily have a constant temperature, but rather a constant energy.

3. What are the equations used to describe adiabatic processes?

The equations used to describe adiabatic processes are the first and second laws of thermodynamics, as well as the ideal gas law. These equations allow us to calculate changes in temperature, pressure, volume, and energy during an adiabatic process.

4. What are some real-life examples of adiabatic processes?

Some real-life examples of adiabatic processes include the compression and expansion of gases in a car engine, the compression of air in a bicycle pump, and the expansion of air in a hot air balloon. These processes are all adiabatic because there is no exchange of heat with the surroundings.

5. Why is a mathematically rigorous treatment of adiabatic processes important?

A mathematically rigorous treatment of adiabatic processes is important because it allows us to accurately predict and analyze the behavior of systems that undergo adiabatic processes. This can be useful in fields such as thermodynamics, engineering, and atmospheric science.

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