Why are 3 states sufficient to describe a statistical ensemble?

In summary, the three variables in the ensembles are based on the different ways we can add energy to a system, and the Legendre transforms allow us to transform the original energy expression into one with variables we can control. A good reference for these topics is Callen's Thermodynamics.
  • #1
blue2script
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Hello!

I have a question arising from my course in statistical physics: Describing microcanonical, canonical and grandcanonical ensemble you take three variables (like E,V,N for microncanonical) and get the "rest" from derivatives of the free energies. Is there a deeper meaning why there are only three variables to be fixed and not four or five?

Another question: Is there any deeper meaning that all free energies transform into each other by Legendre transformations? There should be. Any references on these topics that could be helpful?

Thanks for all answers and help!

Blue2script
 
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  • #2
There are three variables because you are considering increases in system energy via (1) heating, (2) PV work, and (3) adding material. If you also considered adding energy through forming a surface ([itex]\gamma\,dA[/itex]), applying stress ([itex]\sigma V\,d\epsilon[/itex]), etc., you would use more variables (in these cases, [itex]A[/itex] and [tex]\inline{\epsilon}[/tex]) and you'd have a wider number of possible ensembles to try.

As for the potentials question, we always want to use a potential that makes calculations easiest. We know that energy U is minimized for all spontaneous processes, so that [itex]dU=T\,dS-P\,dV+\mu\,dN=0[/itex]. Since it's easy to perform experiments at constant pressure and temperature, let's transform to [itex]G=U-TS+PV[/itex] so that now [itex]dG=-S\,dT+V\,dP+\mu\,dN=0[/itex]. From this we get, for example, that when two phases are in equilibrium the chemical potential [itex]\mu_i[/itex] of material [itex]i[/itex] must be equal in each phase.

A excellent reference is Callen's Thermodynamics.
 
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  • #3
Hi Mapes,

thanks for your answer, it really makes sense! On the second question: As far as I knew are the free energies sort of the energies we can gain when we variate the systems state variables. E.g. G(T,P,N), change T and the difference of both Gibbs energies is the energy you can get out of the system. Is that right? And how is that related to Legendre transforms?

Again, thanks for your answers!

Blue2script
 
  • #4
The reduction in the Gibbs free energy G is the driving force on any process at constant T and P. We can indeed completely capture and store this [itex]\Delta G[/itex] as energy by using a non-heat engine, for example an electrochemical cell, whose efficiency can in theory reach 100% (recall that the efficiency of a heat engine must always be less than 100%).

The substitutions [itex]G=U-TS+PV[/itex], [itex]F=U-TS[/itex], [itex]H=U+PV[/itex], etc. are the Legendre transforms. Their primary use is to transform the original [itex]dU=T\,dS-P\,dV+\mu\,dN+\dots[/itex] into expressions of variables we can control, like T and P. We can extend this idea: if we need to incorporate stress and strain, then we have

[tex]dU=T\,dS-P\,dV+\sigma V\,d\epsilon[/tex]

where I have assumed a closed system so that [itex]dN=0[/itex]. Now let's say I can control temperature, pressure, and stress. Then I define a potential [itex]\phi[/itex] using the Legendre transform

[tex]\phi=U-TS+PV-\sigma V\epsilon[/tex]

which is now minimized when the system is at equilibrium: [itex]d\phi=-S\,dt+V\,dP-\epsilon \,d(\sigma V)=0[/itex]. (If you are familiar with the Clausius-Clapeyron relation, you can probably see how I might change the temperature of a phase transformation by altering the stress on a solid, rather than the pressure which is usually considered. We can calculate things like that with our new potential [itex]\phi[/itex].)
 
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Related to Why are 3 states sufficient to describe a statistical ensemble?

1. Why are only 3 states needed to describe a statistical ensemble?

In statistical mechanics, an ensemble is a collection of similar systems that are in equilibrium. The state of each system can be described by a set of variables, such as energy, temperature, and pressure. These variables are known as the macroscopic state variables. According to the ergodic hypothesis, all possible microstates (i.e. arrangements of particles) that correspond to a given macroscopic state are equally probable. Therefore, in order to fully describe the ensemble, we only need to consider the average values of the macroscopic state variables over all the microstates. This results in a reduction of the number of required states to only 3, as the average values of energy, temperature, and pressure can fully describe the ensemble.

2. Can more than 3 states be used to describe an ensemble?

Yes, it is possible to use more than 3 states to describe an ensemble. In fact, in some cases, it may be necessary to use more states to accurately describe a system. For example, in the case of a system with complex interactions and multiple phases, more than 3 states may be needed. However, in most cases, using only 3 states is sufficient and provides a simpler and more efficient approach.

3. How does using 3 states simplify the description of an ensemble?

Using only 3 states reduces the complexity of describing an ensemble by eliminating the need to consider all possible microstates. Instead, we can focus on the average values of the macroscopic state variables, which are easier to measure and calculate. This simplification also allows for the use of powerful mathematical techniques, such as the canonical ensemble and the grand canonical ensemble, which further simplify the description of ensembles.

4. Are 3 states always sufficient to describe a statistical ensemble?

No, there are cases where 3 states may not be sufficient to fully describe an ensemble. For example, in systems with long-range interactions or in the presence of external fields, the ergodic hypothesis may not hold true and using only 3 states may not accurately describe the system. In these cases, a more detailed description using additional states may be necessary.

5. How does the number of states needed to describe an ensemble relate to the size of the system?

The number of states needed to describe an ensemble is not directly related to the size of the system. In fact, even for a very large system, the use of only 3 states may still be sufficient, as long as the system is in equilibrium and the ergodic hypothesis holds true. However, for larger systems with more complex interactions, it may be necessary to use more states to accurately describe the ensemble.

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