Statistical physics: Microcanonical distribution and oscillators

Thread starter Marcustryi
Start date May 25, 2024
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Hamilton Oscillators

In summary, the microcanonical distribution is a fundamental concept in statistical physics that describes an isolated system with fixed energy, particle number, and volume. It emphasizes the equal probability of all accessible microstates, leading to the derivation of thermodynamic properties. In the context of oscillators, the microcanonical ensemble can be applied to systems of harmonic oscillators, providing insights into energy distribution among modes and the resulting temperature dependence. This framework helps understand the behavior of systems at equilibrium and the statistical nature of thermodynamic quantities.

May 25, 2024

Marcustryi

Homework Statement: An ideal gas, consisting of N harmonic oscillators of mass m and frequency w and charge q, is placed in an external uniform electric field with intensity e. Find the phase volume limited by the energy hypersurface H(x, p; e)=E. Find an expression for the temperature of the oscillator system.

Relevant Equations: H(x, p; e)=E

Hamilton's function in this case is the sum of potential and kinetic energy? But then I don’t remember or don’t understand what to do with e.
I need to find Г, but I don't understand what to do with the field.

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May 27, 2024

Marcustryi

H(x,p;e)=sum(for i=1 to N)(p^2/2m+(mw^2)*x^2/2+qex)=E
Гn=(Г1)^N.
But how to calculate the phase volume? what replacement should I make?

FAQ: Statistical physics: Microcanonical distribution and oscillators

What is the microcanonical ensemble in statistical physics?

The microcanonical ensemble is a statistical ensemble that represents a closed system with fixed energy, volume, and number of particles. It is used to describe isolated systems where all accessible microstates have the same energy. In this ensemble, each microstate is equally probable, and the thermodynamic properties can be derived from the density of states, which counts the number of microstates corresponding to a given energy level.

How do oscillators fit into the microcanonical distribution?

Oscillators are often used as models in statistical physics to represent particles in a system. In the context of the microcanonical distribution, a system of oscillators can be analyzed by considering the energy levels and the number of ways to distribute a fixed amount of energy among these oscillators. The microcanonical ensemble helps us understand the statistical behavior of these oscillators by focusing on the configurations that correspond to a specific energy.

What is the significance of the density of states in the microcanonical ensemble?

The density of states is a crucial concept in the microcanonical ensemble as it quantifies the number of microstates available to the system at a given energy level. It plays a vital role in calculating thermodynamic quantities such as entropy, which is related to the logarithm of the density of states. A higher density of states at a certain energy indicates that there are more ways for the system to achieve that energy, leading to higher entropy.

How does the microcanonical ensemble relate to the concept of entropy?

In the microcanonical ensemble, entropy is defined using Boltzmann's entropy formula, S = k_B * ln(Ω), where S is the entropy, k_B is the Boltzmann constant, and Ω is the number of accessible microstates at a given energy. The microcanonical ensemble allows us to connect the microscopic behavior of particles (through microstates) to macroscopic thermodynamic properties (like entropy), providing a statistical foundation for the second law of thermodynamics.

Can the microcanonical ensemble be applied to real physical systems?

Yes, the microcanonical ensemble can be applied to real physical systems, particularly in cases where the systems can be considered isolated and where energy fluctuations are negligible. It is particularly useful in studying small systems or systems in equilibrium at fixed energy. However, for larger systems or those in contact with a heat bath, other ensembles like the canonical or grand canonical ensembles may be more appropriate.