# Can Entropy be considered a kind of Affine Parameter?

In general relativity and differential geometry, an affine parameter is used to parameterize geodesics such that the geodesic equation $$\nabla_{\mathbf{T}} \mathbf{T} = 0$$ holds true, and the equations describing motion along the geodesic are linear and preserve their form.

In thermodynamics, entropy is a state function that measures the degree of disorder or the number of microstates of a system. According to the second law of thermodynamics, the entropy of an isolated system tends to increase, reflecting the irreversibility and natural tendency towards disorder.

Given these definitions, I am curious whether entropy can be considered a kind of affine parameter in any meaningful sense. Specifically:

1. Can entropy be thought of as measuring "progress" in a manner similar to how an affine parameter measures progress along a geodesic?
2. Is there any context in which the role of entropy in thermodynamic processes could be analogous to the role of an affine parameter in the context of geodesic motion?
3. Are there any physical theories or interpretations where entropy and affine parameters are treated similarly or have analogous functions?

I understand that entropy and affine parameters are used in very different contexts (thermodynamics vs. differential geometry/general relativity), but I am interested in exploring any conceptual or mathematical similarities that might exist between these two ideas.

• Just FYI, the geodesic equation isn't linear, even if you use an affine parameter. Commented Jul 27 at 11:06