Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solution to the Fokker-Planck equation for slowly driven Brownian
  motion: Emergent geometry and a formula for the corresponding thermodynamic
  metric   Considerable progress has recently been made with geometrical approaches to
understanding and controlling small out-of-equilibrium systems, but a
mathematically rigorous foundation for these methods has been lacking. Towards
this end, we develop a perturbative solution to the Fokker-Planck equation for
one-dimensional driven Brownian motion in the overdamped limit enabled by the
spectral properties of the corresponding single-particle Schr\"odinger
operator. The perturbation theory is in powers of the inverse characteristic
timescale of variation of the fastest varying control parameter, measured in
units of the system timescale, which is set by the smallest eigenvalue of the
corresponding Schr\"odinger operator. It applies to any Brownian system for
which the Schr\"odinger operator has a confining potential. We use the theory
to rigorously derive an exact formula for a Riemannian "thermodynamic" metric
in the space of control parameters of the system. We show that up to
second-order terms in the perturbation theory, optimal dissipation-minimizing
driving protocols minimize the length defined by this metric. We also show that
a previously proposed metric is calculable from our exact formula with
corrections that are exponentially suppressed in a characteristic length scale.
We illustrate our formula using the two-dimensional example of a harmonic
oscillator with time-dependent spring constant in a time-dependent electric
field. Lastly, we demonstrate that the Riemannian geometric structure of the
optimal control problem is emergent; it derives from the form of the
perturbative expansion for the probability density and persists to all orders
of the expansion.

17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg