Coarse and fine grained entropy?

[SW VandeCarr]

I'm confused by coarse and fine grained entropy and how they're related to the standard Boltzmann and Gibbs equations. I understand the Gibbs equation is useful even when a system is not in equilibrium. The linked reference seems to say that fine grained entropy is always (or tends to) zero and coarse grained entropy is always (or tends to)positive infinity. Is this true? If so, does it obviate the standard Boltzmann and Gibbs models? Since practical applications are concerned with changes in entropy, not the absolute values, I must not have correctly understand the article. I'm familiar with Shannon entropy, but not so much with the details of thermodynamic entropy.

http://www.mi.ras.ru/~vvkozlov/fulltext/196_e.pdf

 

[SW VandeCarr]

I'll make my question more specific. In section 2 of the above cited paper "..the absence of approximation" the authors state that coarse grained entropy does not approximate fine grained entropy, in the general case, as the partition is refined (last paragraph of section 2). Is this the prevailing view at the present time?

 

[astrokreng]

Coarse and fine grained entropy are terms used to describe different levels of detail in the analysis of a thermodynamic system. In the context of the Boltzmann and Gibbs equations, coarse grained entropy refers to the overall entropy of a macroscopic system, while fine grained entropy refers to the entropy at a microscopic level.

The Boltzmann and Gibbs equations are fundamental equations in statistical mechanics that relate the macroscopic properties of a system, such as temperature and pressure, to the microscopic behaviors of its individual particles. These equations are based on the assumption that the system is in equilibrium, meaning that its macroscopic properties are constant over time.

However, in real-world systems, equilibrium is not always achieved. This is where the Gibbs equation becomes useful, as it allows for the analysis of systems that are not in equilibrium. In this case, the coarse grained entropy may still be defined, but the fine grained entropy may not be well-defined or may tend towards zero.

The statement that coarse grained entropy tends towards positive infinity and fine grained entropy tends towards zero is not necessarily true in all cases. It is important to note that entropy is a property that depends on the level of detail at which it is measured. In other words, the value of entropy can vary depending on the level of granularity at which it is observed.

In terms of practical applications, changes in entropy are what is typically of interest, rather than absolute values. The Boltzmann and Gibbs equations are still valid for analyzing these changes, even when the system is not in equilibrium. However, in these cases, the values of coarse and fine grained entropy may not have a direct physical interpretation.

Overall, the concept of coarse and fine grained entropy is a useful tool for understanding the behavior of thermodynamic systems, particularly in cases where equilibrium is not achieved. However, it is important to keep in mind that entropy is a complex and multifaceted concept that requires careful consideration of the level of detail at which it is measured.