Mircocanonical Damped Harmonic Oscillator

In summary, a microcanonical damped harmonic oscillator is a physical system consisting of a mass attached to a spring and subject to damping forces. It exhibits oscillatory motion with decreasing amplitude due to the damping forces, and is described by the equation of motion <em>m</em> &times; <em>x</em>'' + <em>b</em> &times; <em>x</em>' + <em>k</em> &times; <em>x</em> = 0. The behavior of the system is affected by factors such as mass, spring constant, and damping coefficient, as well as initial conditions. Energy is conserved in the system, with exchanges between kinetic and potential energy and dissipation due to damping
  • #1
jarvGrad
3
0
I am supposed to find the number of mircostates for the following Hamiltonian

[tex]\
\begin{equation}
\Sigma {(q_n+mwp_n)^2}<2mE
\end{equation}
[/tex]

So I am attempting to take the integral as follows

[tex]\
\int e^{(q_n+mwp_n)^2} d^{3n}q d^{3n} p

[tex\]

I found a solution that tells me

[tex]\
\begin{equation}
\int \exp{(ax^2+bxy+cx^2)^2} d^{3n}x d^{3n}y
\end{equation}

[/tex]

which equals

[tex]\
\begin{equation}
\pi^{m/2}/{det[A]}
\end{equation}
[/tex]

where A is the 2-D matrix
A=[a b
b c]

However, the determinant is zero so this doesn't work. I found this solution at http://srikant.org/thesis/node13.html .
There is a bit more work shown on the website. My professor assured me that the solution is closed form.
PS.
Sorry for the bad latex...
This is due tomorrow. So if you could help me fast that would be great!
 
Last edited:
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  • #2

Thank you for sharing your problem with us. I understand the importance of finding a solution quickly, especially when a deadline is approaching. After reviewing your post and the provided link, I have some suggestions that may help you find a closed form solution for your integral.

Firstly, I would like to point out that the integral you are trying to evaluate is a multidimensional Gaussian integral. In general, these types of integrals do not have a closed form solution. However, there are certain cases where they can be evaluated using known techniques.

In your case, the integral can be rewritten as follows:

[tex]
\begin{equation}
\int \exp{[(q_n^2+p_n^2)+2mwp_nq_n]} d^{3n}q d^{3n}p
\end{equation}

This form of the integral may be easier to evaluate. Additionally, you can try to rewrite the exponential term as a product of exponentials, which may help in simplifying the integral.

Moreover, the determinant of the 2-D matrix A is not always zero. It depends on the values of a, b, and c. In fact, if a and c are positive and b is not equal to zero, then the determinant will be positive and the integral will have a solution. You can check this by plugging in different values for a, b, and c.

Lastly, I recommend consulting with your professor or a colleague for further guidance and assistance. They may be able to provide you with additional resources or techniques to help you solve the integral.

I hope this helps and wish you the best of luck in finding a solution. Remember, as scientists, we must persevere through challenges and continue to seek solutions until we find them.
 

1. What is a microcanonical damped harmonic oscillator?

A microcanonical damped harmonic oscillator is a physical system that consists of a mass attached to a spring and subject to damping forces, such as friction. This system is often used to model the behavior of small particles in a confined space.

2. How does a microcanonical damped harmonic oscillator behave?

A microcanonical damped harmonic oscillator exhibits oscillatory motion, meaning that it moves back and forth around an equilibrium position due to the restoring force of the spring. However, the damping forces cause the amplitude of the oscillations to decrease over time, eventually leading to the system reaching a state of equilibrium.

3. What is the equation of motion for a microcanonical damped harmonic oscillator?

The equation of motion for a microcanonical damped harmonic oscillator is given by m × x'' + b × x' + k × x = 0, where m is the mass, x is the displacement from equilibrium, b is the damping coefficient, and k is the spring constant.

4. What factors affect the behavior of a microcanonical damped harmonic oscillator?

The behavior of a microcanonical damped harmonic oscillator is affected by several factors, including the mass of the system, the spring constant, and the damping coefficient. Additionally, the initial conditions, such as the initial displacement and velocity, also play a role in the behavior of the system.

5. How is the energy conserved in a microcanonical damped harmonic oscillator?

In a microcanonical damped harmonic oscillator, the total energy of the system remains constant, meaning that energy is conserved. The energy is exchanged between kinetic energy, due to the motion of the mass, and potential energy, stored in the spring. The damping forces dissipate some of the energy, but the total energy of the system remains constant.

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