Equilibrium state proof with lagrange

In summary, the conversation is about a person seeking clarification on the steps in a set of lecture notes regarding entropy methods. They are specifically having trouble understanding how the equation e^{1-λ} leads to 1/N, as they believe N should represent the number of micro states. They are asking for someone to explain this to them.
  • #1
Guffie
23
0
Hello,

I have been trying to follow the start of these lecture notes I found online and I have having trouble understanding what is happening between two steps.

The notes I am looking at are located:
http://pillowlab.cps.utexas.edu/teaching/CompNeuro10/slides/slides16_EntropyMethods.pdf [Broken]

On the bottom of slide 4 they get p_j = e^{1-λ} which is clear to me.

The next line says = 1/N.

I am having trouble understanding how they get to 1/N, from my understanding the N should represent the number of micro states however I don't understand how they jump from e^{1-λ} to 1/N.

Would anyone be able to clarify this for me?

Thank you,
 
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  • #2
The probability is the same for all j, so therefore you have [itex]\sum p_i = 1 = N p_i [/itex].
 

1. What is the Equilibrium State Proof with Lagrange?

The Equilibrium State Proof with Lagrange is a method used to prove the existence of an equilibrium state in a dynamic system. It involves using the Lagrange multipliers to find the values of the system's variables that satisfy both the system's equations of motion and the constraints.

2. Why is the Equilibrium State Proof with Lagrange important?

The Equilibrium State Proof with Lagrange is important because it allows us to mathematically prove the existence of an equilibrium state in a dynamic system. This is useful in many fields, such as physics, engineering, and economics, where understanding the behavior of systems at equilibrium is crucial.

3. How does the Equilibrium State Proof with Lagrange work?

The Equilibrium State Proof with Lagrange involves setting up a system of equations that represent the equations of motion and constraints of the system. The Lagrange multipliers are then introduced as additional variables and the equations are solved to find the values of the system's variables that satisfy all of the equations.

4. What are the assumptions made in the Equilibrium State Proof with Lagrange?

The Equilibrium State Proof with Lagrange assumes that the system is in a steady state, meaning that its properties do not change over time. It also assumes that the equations of motion and constraints are continuous and differentiable, and that the Lagrange multipliers are non-negative.

5. Can the Equilibrium State Proof with Lagrange be applied to all systems?

The Equilibrium State Proof with Lagrange can be applied to systems that meet the assumptions mentioned above. However, it may not always be the most efficient or practical method for proving the existence of an equilibrium state in a system. Other methods, such as graphical or numerical analysis, may be more suitable depending on the specific system and its properties.

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