Thermodynamics Boltzmann Statistics

In summary, the conversation discusses a classical particle in a potential well and its average position in thermal equilibrium with a reservoir at temperature T. The average position is shown to be determined by Boltzmann statistics and can be calculated using integrals and the partition function. The use of classical ensembles is also mentioned.
  • #1
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Homework Statement


Consider a classical particle moving in a one-dimensional potential well u(x). The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics. Show that the average position of the particle is given by [tex]\overline{x}=\frac{\int xe^{-(\beta)u(x)}\,dx}{\int e^{-(\beta)u(x)}\,dx}[/tex]

Homework Equations


Partition function, equipartition theorem

The Attempt at a Solution


I don't know where they get the integrals from, the partition function is a sum.
 
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  • #2
It's a sum over all possible states. In the classical case, the states of a system is determined by the positions and momenta of all particles in the system. But position and momenta are continuous variables, so the sum becomes an integral.
 
  • #3
He says right there: "classical particle". So you have to use the "classical" ensembles, namely the canonic one.

Daniel.
 
  • #4
Do you ever look at a problem for a really long time, not having any clue how to do it, and then all of a sudden something just clicks? This is one of those problems.
 

1. What is the basic concept of thermodynamics Boltzmann statistics?

Thermodynamics Boltzmann statistics is a branch of thermodynamics that studies the behavior of a large number of particles in a system by using statistical methods. It is based on the idea that the state of a system can be described by the distribution of energy among its particles.

2. How does Boltzmann statistics relate to the laws of thermodynamics?

Boltzmann statistics is based on the laws of thermodynamics, which state that energy cannot be created or destroyed, only transferred or converted. Boltzmann statistics explains how energy is distributed among particles in a system, which is essential for understanding thermodynamic processes.

3. What is the Boltzmann distribution and how is it used in thermodynamics?

The Boltzmann distribution is a mathematical equation that describes the probability of a particle having a certain amount of energy in a system at a given temperature. It is used in thermodynamics to calculate the most probable distribution of energy among particles in a system, which can then be used to determine the thermodynamic properties of the system.

4. What is the significance of Boltzmann's constant in thermodynamics Boltzmann statistics?

Boltzmann's constant (k) is a fundamental constant in thermodynamics Boltzmann statistics that relates the energy of a particle to its temperature. It is used to convert between the microscopic scale (individual particles) and the macroscopic scale (bulk properties of a system).

5. What are some practical applications of thermodynamics Boltzmann statistics?

Thermodynamics Boltzmann statistics has many practical applications, such as in the design of refrigeration and heat exchange systems, the study of chemical reactions, and the analysis of gas behavior. It is also used in various fields of physics, chemistry, and engineering to understand and predict the behavior of complex systems.

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