Partition function in polymers and entropic forces

[ianhoolihan]

Hi all,

Firstly, I'm not sure where to post this thread, but I'm hoping here is the right place. My questions developed through reading Verlinde's paper on entropic gravity:

http://arxiv.org/abs/1001.0785"

However my questions are with the introductory thermodynamic ideas he presents on entropy with polymers.

Firstly, I was wondering how the partition function (2.2) was arrived at? Specifically, from all that I've read on the internet, I don't see where the omega factor comes in?

Secondly, how are the relations (2.3) derived from the 'saddle point equations'? From what I gather, solving these saddle-point equations (2.3) gives the saddle points of the function $$\Omega{}e^{-(E+Fx)/kt}$$. I should then be able to evaluate the integral Z. But why do I want to do that? What is the signficance of these saddle point equations? In terms of entropy?

Thirdly, how does (2.4) imply these saddle equations? I can see how if one defines $$\frac{dS}{dE}=\frac{1}{T}$$, but he seems to indicate that this is not needed?

Anyway, any help would be much appreciated = )

Cheers

 

[eljose79]

Hi there,

Thank you for your questions about Verlinde's paper on entropic gravity. I can offer some insights into the concepts and equations you are asking about.

Firstly, the partition function (2.2) is a fundamental concept in statistical mechanics, which is a branch of physics that deals with the behavior of large systems of particles. In this context, the partition function represents the total number of ways that a system can be arranged or distributed among its possible energy states. The omega factor, also known as the degeneracy factor, takes into account that some energy states may have multiple configurations or arrangements. It is derived from the combinatorial principle, which states that the number of ways to arrange n objects in a certain number of ways is equal to n!. In the case of polymers, the omega factor takes into account the different ways that the polymer chains can be arranged in space.

Secondly, the saddle point equations (2.3) are derived from the principle of maximum entropy, also known as the maximum entropy principle. This principle states that a system will tend towards the state with the highest entropy, or the most disordered state. In the context of polymers, this means that the system will tend towards the state with the most possible arrangements of the polymer chains. The saddle point equations are used to find the maximum of the function \Omega{}e^{-(E+Fx)/kt}, which represents the total number of arrangements of the polymer chains at a given energy and force. This maximum corresponds to the most probable state of the system, which can then be used to evaluate the partition function Z.

Thirdly, equation (2.4) is derived from the fundamental thermodynamic relation \frac{dS}{dE}=\frac{1}{T}, which relates the change in entropy to the change in energy at constant temperature. In the context of polymers, this means that the change in entropy is equal to the change in energy divided by the temperature. This relation is not explicitly needed to derive the saddle point equations, but it can be used to interpret the meaning of the saddle point equations in terms of entropy.

I hope this helps to clarify some of the concepts in Verlinde's paper. Keep in mind that these are complex and abstract ideas, and it may take some time and effort to fully understand them. Don't hesitate to continue asking questions and seeking out resources to deepen your understanding. Good luck with your