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I edited the paragraphs on the justification of the microcanonical ensemble, to include the fact that chaotic dynamics (in addition to ergodicity) are also necessary to fully justify the microcanonical description.
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Wade Hodson
Wade Hodson
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I think there are two related questions here.

First, we might simply assume that the microcanonical ensemble and its associated probability distribution offer an appropriate description of the joint system + bath. We can then ask the question: Under what circumstances will the system be well-described by a canonical ensemble? The general answer to this question is that the joint system must satisfy two conditions:

  1. The bath should be much larger than the system; that is, it should have many more degrees of freedom than the system. This implies, among other things, that the energy of the bath will be much greater than that of the system (at least for a typical sample from the microcanonical distribution of the joint system).

  2. The interaction Hamiltonian $H_{interaction}$ should be small in some sense relative to the system Hamiltonian $H_{system}$.

These conditions may sound a bit imprecise. However, at this level of generality, where the characteristics of the system and bath are left mostly unspecified, I don't think that that's necessarily a bad thing. However, if we have a specific joint system in mind, then we can always try to flesh out these conditions in more detail.

Now, if (1) and (2) hold, and we assume that the joint system is describe by a microcanonical ensemble, then standard arguments found in intro stat mech textbooks can be used to show that the system may be described by a canonical ensemble.

However, this isn't the whole story. We can ask another question: When are we justified in using the microcanonical ensemble to describe the joint system in the first place? For classical Hamiltonian systems, the standard answer is that the joint system must be ergodic and chaotic, or approximately ergodic and chaotic in some sense. Roughly, thisergodicity means that over the course of its evolution, the joint system visits every state available to it consistent with its total energy being constant (more precisely, it comes arbitrarily close to all such states), and the term chaotic refers to a sensitive dependence of the joint system's evolution on its initial state.

It's difficult to prove that a given system is ergodic and chaotic. To my knowledge, certain classes of dynamical billiards are some of the few types of systems which have been rigorously proven to be ergodic and chaotic. However, what is clear is that conditions (1) and (2) alone aren't enough to establish chaos or ergodicity (and therefore to justify the microcanonical ensemble). For example, we can satisfy (1) and (2) with a joint system with $H_{interaction}=0$. For such a joint system, the system and bath will be noninteracting, and so the individual energies of the system and the bath will each be constant. This joint system is not ergodic, since it will only visit states where the system and bath energies separately remain unchanged.

In summary, if we assume the validity of the microcanonical description for the joint system, then conditions (1) and (2) ensure that the system will be described by a canonical ensemble. However, to justify our microcanonical assumption, we need to further establish that the joint system is ergodic and chaotic, which is a nontrivial task to say the least.

I think there are two related questions here.

First, we might simply assume that the microcanonical ensemble and its associated probability distribution offer an appropriate description of the joint system + bath. We can then ask the question: Under what circumstances will the system be well-described by a canonical ensemble? The general answer to this question is that the joint system must satisfy two conditions:

  1. The bath should be much larger than the system; that is, it should have many more degrees of freedom than the system. This implies, among other things, that the energy of the bath will be much greater than that of the system (at least for a typical sample from the microcanonical distribution of the joint system).

  2. The interaction Hamiltonian $H_{interaction}$ should be small in some sense relative to the system Hamiltonian $H_{system}$.

These conditions may sound a bit imprecise. However, at this level of generality, where the characteristics of the system and bath are left mostly unspecified, I don't think that that's necessarily a bad thing. However, if we have a specific joint system in mind, then we can always try to flesh out these conditions in more detail.

Now, if (1) and (2) hold, and we assume that the joint system is describe by a microcanonical ensemble, then standard arguments found in intro stat mech textbooks can be used to show that the system may be described by a canonical ensemble.

However, this isn't the whole story. We can ask another question: When are we justified in using the microcanonical ensemble to describe the joint system in the first place? For classical Hamiltonian systems, the standard answer is that the joint system must be ergodic, or approximately ergodic in some sense. Roughly, this means that over the course of its evolution, the joint system visits every state available to it consistent with its total energy being constant (more precisely, it comes arbitrarily close to all such states).

It's difficult to prove that a given system is ergodic. To my knowledge, certain classes of dynamical billiards are some of the few types of systems which have been rigorously proven to be ergodic. However, what is clear is that conditions (1) and (2) alone aren't enough to establish ergodicity (and therefore to justify the microcanonical ensemble). For example, we can satisfy (1) and (2) with a joint system with $H_{interaction}=0$. For such a joint system, the system and bath will be noninteracting, and so the individual energies of the system and the bath will each be constant. This joint system is not ergodic, since it will only visit states where the system and bath energies separately remain unchanged.

In summary, if we assume the validity of the microcanonical description for the joint system, then conditions (1) and (2) ensure that the system will be described by a canonical ensemble. However, to justify our microcanonical assumption, we need to further establish that the joint system is ergodic, which is a nontrivial task to say the least.

I think there are two related questions here.

First, we might simply assume that the microcanonical ensemble and its associated probability distribution offer an appropriate description of the joint system + bath. We can then ask the question: Under what circumstances will the system be well-described by a canonical ensemble? The general answer to this question is that the joint system must satisfy two conditions:

  1. The bath should be much larger than the system; that is, it should have many more degrees of freedom than the system. This implies, among other things, that the energy of the bath will be much greater than that of the system (at least for a typical sample from the microcanonical distribution of the joint system).

  2. The interaction Hamiltonian $H_{interaction}$ should be small in some sense relative to the system Hamiltonian $H_{system}$.

These conditions may sound a bit imprecise. However, at this level of generality, where the characteristics of the system and bath are left mostly unspecified, I don't think that that's necessarily a bad thing. However, if we have a specific joint system in mind, then we can always try to flesh out these conditions in more detail.

Now, if (1) and (2) hold, and we assume that the joint system is describe by a microcanonical ensemble, then standard arguments found in intro stat mech textbooks can be used to show that the system may be described by a canonical ensemble.

However, this isn't the whole story. We can ask another question: When are we justified in using the microcanonical ensemble to describe the joint system in the first place? For classical Hamiltonian systems, the standard answer is that the joint system must be ergodic and chaotic, or approximately ergodic and chaotic in some sense. Roughly, ergodicity means that over the course of its evolution, the joint system visits every state available to it consistent with its total energy being constant (more precisely, it comes arbitrarily close to all such states), and the term chaotic refers to a sensitive dependence of the joint system's evolution on its initial state.

It's difficult to prove that a given system is ergodic and chaotic. To my knowledge, certain classes of dynamical billiards are some of the few types of systems which have been rigorously proven to be ergodic and chaotic. However, what is clear is that conditions (1) and (2) alone aren't enough to establish chaos or ergodicity (and therefore to justify the microcanonical ensemble). For example, we can satisfy (1) and (2) with a joint system with $H_{interaction}=0$. For such a joint system, the system and bath will be noninteracting, and so the individual energies of the system and the bath will each be constant. This joint system is not ergodic, since it will only visit states where the system and bath energies separately remain unchanged.

In summary, if we assume the validity of the microcanonical description for the joint system, then conditions (1) and (2) ensure that the system will be described by a canonical ensemble. However, to justify our microcanonical assumption, we need to further establish that the joint system is ergodic and chaotic, which is a nontrivial task to say the least.

Source Link
Wade Hodson
Wade Hodson
  • 674
  • 5
  • 13

I think there are two related questions here.

First, we might simply assume that the microcanonical ensemble and its associated probability distribution offer an appropriate description of the joint system + bath. We can then ask the question: Under what circumstances will the system be well-described by a canonical ensemble? The general answer to this question is that the joint system must satisfy two conditions:

  1. The bath should be much larger than the system; that is, it should have many more degrees of freedom than the system. This implies, among other things, that the energy of the bath will be much greater than that of the system (at least for a typical sample from the microcanonical distribution of the joint system).

  2. The interaction Hamiltonian $H_{interaction}$ should be small in some sense relative to the system Hamiltonian $H_{system}$.

These conditions may sound a bit imprecise. However, at this level of generality, where the characteristics of the system and bath are left mostly unspecified, I don't think that that's necessarily a bad thing. However, if we have a specific joint system in mind, then we can always try to flesh out these conditions in more detail.

Now, if (1) and (2) hold, and we assume that the joint system is describe by a microcanonical ensemble, then standard arguments found in intro stat mech textbooks can be used to show that the system may be described by a canonical ensemble.

However, this isn't the whole story. We can ask another question: When are we justified in using the microcanonical ensemble to describe the joint system in the first place? For classical Hamiltonian systems, the standard answer is that the joint system must be ergodic, or approximately ergodic in some sense. Roughly, this means that over the course of its evolution, the joint system visits every state available to it consistent with its total energy being constant (more precisely, it comes arbitrarily close to all such states).

It's difficult to prove that a given system is ergodic. To my knowledge, certain classes of dynamical billiards are some of the few types of systems which have been rigorously proven to be ergodic. However, what is clear is that conditions (1) and (2) alone aren't enough to establish ergodicity (and therefore to justify the microcanonical ensemble). For example, we can satisfy (1) and (2) with a joint system with $H_{interaction}=0$. For such a joint system, the system and bath will be noninteracting, and so the individual energies of the system and the bath will each be constant. This joint system is not ergodic, since it will only visit states where the system and bath energies separately remain unchanged.

In summary, if we assume the validity of the microcanonical description for the joint system, then conditions (1) and (2) ensure that the system will be described by a canonical ensemble. However, to justify our microcanonical assumption, we need to further establish that the joint system is ergodic, which is a nontrivial task to say the least.