# Statistical mechanics question

• bon
In summary, the conversation discusses the concept of a population inversion in a two level system. This is a situation where there are more electrons in the higher energy state than the lower energy state, which is not an equilibrium situation. The ratio of populations in the two states can still be expressed by an effective temperature. It is shown that for a population inversion to exist, the effective temperature must be negative. If the populations were to be swapped, the new effective temperature can be determined. Finally, the effective temperature if all the electrons were in the upper state is discussed. The solution to this problem can be found using the Gibbs distribution and solving a 2 state system at a given temperature.

## Homework Statement

Suppose that by some artificial means it is possible to put more electrons in the higher energy state than in the lower energy state of a two level system. Now it is clear that this system cannot be an equilibrium situation, but, nevertheless, for the time that the system is in this strange state we could, if we wished, still express the ratio of the populations in the upper and lower states by some parameter we can think of as an effective temperature.

(i) show that for such a population inversion to exist, the effective temperature must be negative

(ii) imagine that i have electrons that populate the two states in the normal manner at room temperature. I then somehow swap the populations (i/e/ all the ones that were in thw lower temperature go into the upper state, and vice versa) What is the new effective temperature?

(iii) what is the effective temperature if I put all the electrons in the upper state?

## The Attempt at a Solution

not sure where to begin! any help would be great. thanks

Use Gibbs distribution.

Begin by solving a 2 state system at some temperature T. You may do this in the microcanonical ensemble or more easily, in the canonical ensemble.

You should find that, if the states are separated by an energy E, that the population in the higher energy state is

1/(1 + exp(-E/kT)).

You should be able to handle it from there.

## 1. What is statistical mechanics?

Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of large systems of particles, such as atoms and molecules. It aims to understand how macroscopic properties of a system, such as temperature and pressure, arise from the microscopic behavior of its constituent particles.

## 2. What are the key principles of statistical mechanics?

The key principles of statistical mechanics include the assumption of equal a priori probabilities, the ergodic hypothesis, and the use of statistical ensembles to describe the behavior of a system. These principles allow for the calculation of thermodynamic quantities, such as entropy and free energy, from the microscopic properties of a system.

## 3. How is statistical mechanics related to thermodynamics?

Thermodynamics and statistical mechanics are closely related, with statistical mechanics providing a microscopic understanding of the macroscopic laws of thermodynamics. While thermodynamics describes the behavior of macroscopic systems in equilibrium, statistical mechanics provides a more detailed understanding of the underlying microscopic processes that lead to thermodynamic behavior.

## 4. What types of systems can be studied using statistical mechanics?

Statistical mechanics can be applied to a wide range of systems, from simple ideal gases to complex systems such as liquids, solids, and even biological systems. It is particularly useful for systems with a large number of particles, where traditional analytical methods may be difficult to apply.

## 5. What are some practical applications of statistical mechanics?

Statistical mechanics has many practical applications, including the design of materials with specific properties, the study of phase transitions, and the understanding of complex biological systems. It is also used in fields such as chemistry, engineering, and materials science to model and predict the behavior of physical systems at the microscopic level.

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