Equivalent descriptions of the loewner energy

Wang, Yilin

doi:10.1007/s00222-019-00887-0

Cited by 38 publications

(52 citation statements)

References 47 publications

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“…Although non-equilibrium systems typically lack the conserved quantities, such as the Hamiltonian in the equilibrium physics, our result indicates that the entropy (or associated energy function) of the driving function is conserved, even if the corresponding curve retains the non-equilibrium state. A similar type of perspective was suggested in the mathematical context [ 56 , 57 ], where the invariance of the Dirichlet energy of the Loewner driving function (referred to as the Loewner energy ) was demonstrated. For these reasons, we suggest that entropy of the Loewner driving function may be referred to as the Loewner entropy .…”

Section: Discussionmentioning

confidence: 80%

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

Shibasaki

Saito

2021

Entropy

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In this study, we theoretically investigated a generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics in the context of non-equilibrium statistical mechanics. Using the ability of Loewner evolution, which enables encoding of non-equilibrium systems into equilibrium systems, we formulated the encoding mechanism of the SLE by Gibbs entropy-based information-theoretic approaches to discuss its advantages as a means to better describe non-equilibrium systems. After deriving entropy production and flux for the 2D trajectories of the generalized SLE curves, we reformulated the system’s entropic properties in terms of the Kullback–Leibler (KL) divergence. We demonstrate that this operation leads to alternative expressions of the Jarzynski equality and the second law of thermodynamics, which are consistent with the previously suggested theory of information thermodynamics. The irreversibility of the 2D trajectories is similarly discussed by decomposing the entropy into additive and non-additive parts. We numerically verified the non-equilibrium property of o ur model by simulating the long-time behavior of the entropic measure suggested by our formulation, referred to as the relative Loewner entropy.

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Section: Discussionmentioning

confidence: 80%

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

Shibasaki

Saito

2021

Entropy

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“…It is convenient to start with the bounded case. See [8,27,30,33] for proofs of the results summarized in the next theorem.…”

Section: Loewner Energymentioning

confidence: 99%

“…Let f : H → H and g : H * → H * be conformal maps fixing ∞. The Loewner energy of η can be expressed in terms of f and g as (see [33] Theorem 6.1)…”

Section: )mentioning

confidence: 99%

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Viklund

Wang

2020

Geom. Funct. Anal.

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The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.

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“…It also suggests that loop energy has to be a more fundamental quantity. Indeed, in a later work [24] of very different flavor, the second author derived equivalent descriptions of the loop energy connecting to Weil-Petersson class of universal Teichmüller space.…”

Section: Conventionsmentioning

confidence: 99%

The Loewner Energy of Loops and Regularity of Driving Functions

Rohde

Wang

2019

International Mathematics Research Notices

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Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a C 1,β curve (differentiable parametrization with β-Hölder continuous derivative) is in the class C 1,β−1/2 if 1/2 < β ≤ 1, and in the class C 0,β+1/2 if 0 ≤ β ≤ 1/2. This is the converse of a result of Carto Wong [27] and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. Recently, in [6] and [23] the chordal Loewner energy ∞ 0Ẇ (t) 2 /2 dt of η was introduced independently, and basic properties (such as rectifiability) of curves with finite energy were obtained. The chordal Loewner energy a priori depends on the orientation of η, namely viewed as a curve from 0 to ∞ or from ∞ to 0. However, the second author [23] proved the direction-independence (or reversibility).In this paper, we generalize the definition of Loewner energy to simple loops on the Riemann sphere γ : R →Ĉ where γ is continuous, 1-periodic and injective on [0, 1) : We just observe that the limit when ε → 0 of the chordal energy of γ [ε, 1] in the simply connected domainĈ \ γ[0, ε] exists in [0, ∞] (Proposition 2.8), and define it as the loop Loewner energy of γ rooted at γ(0). Note that circles have loop energy 0. Intuitively, the loop energy measures how much the Jordan curve γ[0, 1] differs from a circle seen from the root γ(0), in a Möbius invariant fashion. The loop Loewner energy generalizes * the chordal Loewner energy: Indeed, if we apply z → z 2 to a chord η from 0 to ∞ in H, the positive real line together with the image of η forms a loop γ through ∞. It is clear that its loop energy rooted at ∞ (i.e. we parametrize the loop such that γ(0) = ∞) equals the chordal energy of η.Note also that the loop energy neither depends on any increasing reparametrization of γ fixing the root, nor on the direction of parametrization. The latter fact basically comes from the chordal reversibility, which can be used to show thatγ(t) = γ(1 − t) has the same energy as γ (see Section 2.2 for details). But it depends a priori on the root γ(0) where the limit is taken, not only on the Jordan curve γ[0, 1]. However, our first main result states:Theorem 1.1. The loop Loewner energy is root-invariant.In particular, this result shows that the loop Loewner energy is a conformal invariant on the set of unrooted loops on the Riemann sphere, which attains its minimum 0 only on circles.In our proof of the root-invariance, we approximate the curve by well-chosen regular curves and are led to the following question: What can we say about the relation between the regularity of the driving function and the regularity of the curve? Prior to this work, only one direction was well understood. Slightly imprecisely, the following results state that C α driving functions generate C α+1/2 curves for α > 1/2, where C α is understood with the usual convention as C n,β , where n is the integer part of α and β =...

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Equivalent descriptions of the loewner energy

Cited by 38 publications

References 47 publications

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

Non-Equilibrium Entropy and Irreversibility in Generalized Stochastic Loewner Evolution from an Information-Theoretic Perspective

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

The Loewner Energy of Loops and Regularity of Driving Functions

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