Small deviations from equilibrium and Lagrange multipliers

In summary, according to "Principles of Statistical Mechanics" by Amnon Katz, for small deviations from equilibrium, the expression ##\exp ( -\alpha N )## must be able to be expanded in powers of ##\alpha## with only the first order term kept. The necessary condition for this is that the statistical function ##f## takes the form ##\exp(\Omega + \alpha N)##, and the Lagrange multiplier ##\alpha## is obtained through the correct derivation.
  • #1
Ted Ali
12
1
Homework Statement
Consider the grand canonical ensemble, for a system A with N identical particles. This system can exchange particles with a reservoir A'. We consider only small deviations from equilibrium. What is the necessary and sufficient condition that the Lagrange multiplier ##\alpha## of N, must satisfy?
Relevant Equations
The Lagrange multiplier ##\alpha## is: $$\alpha =\left( \frac{\partial \ln \Omega}{\partial N'} \right)$$.
According to the book "Principles of Statistical Mechanics" by Amnon Katz, page 123, ##\alpha## must be such that ##\exp ( -\alpha N ) ## can be expanded in powers of ##\alpha## with only the first order term kept. Is this the necessary and sufficient condition for small deviations from equilibrium?

Thank you in advance,
Ted.
 
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  • #2
Hello Ted.

Looking through your expressions, I guess that you are using the same notation of the book ("Principles of Statistical Mechanics" by Amnon Katz).

Two points:

What is the expression of your statistical functions ##f##'s?
I assume that it should be something like this for system ##A##,

$$f=\exp(\Omega\ +\ \alpha N )\ ,$$
right? (please correct me if I am wrong).

May you elaborate on how you obtained the Lagrange multiplayer ##\alpha##? If your derivation is correct, then you have the necessary condition.
 

FAQ: Small deviations from equilibrium and Lagrange multipliers

1. What is equilibrium and how do small deviations from it affect a system?

Equilibrium is a state in which all forces and factors in a system are balanced, resulting in no change or movement. Small deviations from equilibrium occur when there is a slight disturbance or change in the system, causing it to move away from its balanced state. These deviations can have significant impacts on the behavior and stability of a system.

2. What are Lagrange multipliers and how are they used in studying small deviations from equilibrium?

Lagrange multipliers are mathematical tools used to optimize a function subject to constraints. In the context of studying small deviations from equilibrium, they are used to find the optimal conditions for a system to return to equilibrium after a disturbance. They help us understand the relationship between the variables in a system and how they are affected by small changes.

3. How do small deviations from equilibrium contribute to the study of dynamic systems?

Dynamic systems are constantly changing and evolving, and small deviations from equilibrium play a crucial role in this process. By studying these deviations, we can gain a better understanding of how a system responds to external factors and how it maintains stability in the face of disturbances. This knowledge can be applied to various fields such as physics, chemistry, and biology.

4. Can small deviations from equilibrium be beneficial for a system?

Yes, small deviations from equilibrium can have both positive and negative effects on a system. In some cases, they can lead to new and improved states of equilibrium, allowing the system to adapt and evolve. However, if the deviations are too large or frequent, they can cause instability and potentially lead to the collapse of the system.

5. How can the study of small deviations from equilibrium be applied in real-world situations?

The study of small deviations from equilibrium has many practical applications, such as in engineering, economics, and environmental science. By understanding how systems respond to disturbances, we can design more efficient and stable structures, predict and prevent potential crises, and make informed decisions about resource management and conservation.

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