Suppose a set [itex]X[/itex] describes the possible states of some system, and suppose a function [itex]x\mapsto E(x)[/itex] tells the energy level of each state. At temperature [itex]T[/itex] the Boltzmann-measure, which will be the probability measure describing the state of the system, is obtained by formula
[tex]
dp(x) = \frac{1}{Z(T)} e^{-\frac{E(x)}{k_{\textrm{B}}T}} d\mu(x)
[/tex]
where [itex]Z(T)[/itex] has been defined by
[tex]
Z(T) = \int\limits_X e^{-\frac{E(x)}{k_{\textrm{B}}T}} d\mu(x)
[/tex]
and where [itex]\mu[/itex] IS SOME MYSTERIOUS BACKGROUND MEASURE, which seems to avoided in all physics literature.
For example, if we want to derive the Maxwell-Boltzmann distribution for particles in gas, we denote [itex]v=x[/itex] (since the state of the particle is described by its velocity (or momentum) in this model), set [itex]E(v)=\frac{1}{2}m\|v\|^2[/itex] and [itex]\mu=m_3[/itex], where [itex]m_3[/itex] is the ordinary three dimensional Lebesgue-measure.
Another example: In Ising model we have [itex]X=\{-1,+1\}^L[/itex] where [itex]L[/itex] is a set whose elements describe lattice points. (Let's assume that [itex]L[/itex] is finite.) Then we define [itex]\mu[/itex] as the number measure so that [itex]\mu(\{x\})=1[/itex], and for non-trivial [itex]A\subset X[/itex] [itex]\mu(A)[/itex] tells the number of elements (states) in [itex]A[/itex]. The energy function [itex]E[/itex] is defined by using information about which points are neighbour points.
So the Boltzmann-measure consists of two parts. One part is the function [itex]e^{-E/(k_{\textrm{B}}T)}[/itex], and the second part is some background measure. The Boltzmann-measure is obtained, when the background measure is weighted with the function that depends on the energy and temperature.
Everytime I try to read about statistical physics, I only find discussion about the function [itex]e^{-E/(k_{\textrm{B}}T)}[/itex], but not about the background measure.
Suppose I define a measure [itex]\mu[/itex] by a formula [itex]d\mu(x)=(e^{-\|x\|} + \sin(\|x\|))dm_3(x)[/itex], and then claim that my Maxwell-Boltzmann measure is
[tex]
dp(x)\sim e^{-\frac{m\|x\|^2}{2k_{\textrm{B}}T}} d\mu(x)
[/tex]
Why is this wrong? I calculated "the function" correctly, and then weighted "some measure" with "the correct function".
How do we solve the correct background measure, which you will then weight with the function?
I have found this topic to be very difficult and frustrating. Everytime I have attempted to ask about the background measure, people change the topic to the function. Even professors. Like I explain carefully that I have understood where [itex]e^{-E/(k_{\textrm{B}}T)}[/itex] comes from, but I have not understood where [itex]\mu[/itex] comes from. Then people stare at me as if I was dumb and respond "the derivation of [itex]e^{-E/(k_{\textrm{B}}T)}[/itex] was explained right there!". Apparently physicists don't like questions to which they don't know answers?
[tex]
dp(x) = \frac{1}{Z(T)} e^{-\frac{E(x)}{k_{\textrm{B}}T}} d\mu(x)
[/tex]
where [itex]Z(T)[/itex] has been defined by
[tex]
Z(T) = \int\limits_X e^{-\frac{E(x)}{k_{\textrm{B}}T}} d\mu(x)
[/tex]
and where [itex]\mu[/itex] IS SOME MYSTERIOUS BACKGROUND MEASURE, which seems to avoided in all physics literature.
For example, if we want to derive the Maxwell-Boltzmann distribution for particles in gas, we denote [itex]v=x[/itex] (since the state of the particle is described by its velocity (or momentum) in this model), set [itex]E(v)=\frac{1}{2}m\|v\|^2[/itex] and [itex]\mu=m_3[/itex], where [itex]m_3[/itex] is the ordinary three dimensional Lebesgue-measure.
Another example: In Ising model we have [itex]X=\{-1,+1\}^L[/itex] where [itex]L[/itex] is a set whose elements describe lattice points. (Let's assume that [itex]L[/itex] is finite.) Then we define [itex]\mu[/itex] as the number measure so that [itex]\mu(\{x\})=1[/itex], and for non-trivial [itex]A\subset X[/itex] [itex]\mu(A)[/itex] tells the number of elements (states) in [itex]A[/itex]. The energy function [itex]E[/itex] is defined by using information about which points are neighbour points.
So the Boltzmann-measure consists of two parts. One part is the function [itex]e^{-E/(k_{\textrm{B}}T)}[/itex], and the second part is some background measure. The Boltzmann-measure is obtained, when the background measure is weighted with the function that depends on the energy and temperature.
Everytime I try to read about statistical physics, I only find discussion about the function [itex]e^{-E/(k_{\textrm{B}}T)}[/itex], but not about the background measure.
Suppose I define a measure [itex]\mu[/itex] by a formula [itex]d\mu(x)=(e^{-\|x\|} + \sin(\|x\|))dm_3(x)[/itex], and then claim that my Maxwell-Boltzmann measure is
[tex]
dp(x)\sim e^{-\frac{m\|x\|^2}{2k_{\textrm{B}}T}} d\mu(x)
[/tex]
Why is this wrong? I calculated "the function" correctly, and then weighted "some measure" with "the correct function".
How do we solve the correct background measure, which you will then weight with the function?
I have found this topic to be very difficult and frustrating. Everytime I have attempted to ask about the background measure, people change the topic to the function. Even professors. Like I explain carefully that I have understood where [itex]e^{-E/(k_{\textrm{B}}T)}[/itex] comes from, but I have not understood where [itex]\mu[/itex] comes from. Then people stare at me as if I was dumb and respond "the derivation of [itex]e^{-E/(k_{\textrm{B}}T)}[/itex] was explained right there!". Apparently physicists don't like questions to which they don't know answers?