Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Definitions and Evolutions of Statistical Entropy for Hamiltonian
  Systems   Regardless of studies and debates over a century, the statistical origin of
the second law of thermodynamics still remains illusive. One essential obstacle
is the lack of a proper theoretical formalism for non-equilibrium entropy. Here
I revisit the seminal ideas about non-equilibrium statistical entropy due to
Boltzmann and due to Gibbs, and synthesize them into a coherent and precise
framework. Using this framework, I clarify the anthropomorphic principle of
entropy, and analyze the evolution of entropy for classical Hamiltonian systems
under different experimental setups. I find that evolution of Boltzmann entropy
obeys a Stochastic H-Theorem, which relates probability of Boltzmann entropy
increasing to that of decreasing. By contrast, the coarse-grained Gibbs entropy
is monotonically increasing, if the microscopic dynamics is locally mixing, and
the initial state is a Boltzmann state. These results clarify the precise
meaning of the second law of thermodynamics for classical systems, and
demonstrate that it is the initial condition as a Boltzmann state that is
ultimately responsible for the arrow of time.

9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
2
The categories of this paper are: math
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom