Conformal Fisher information metric with torsion Kunal Pal1Kuntal Pal1yand Tapobrata Sarkar1z 1Department of Physics Indian Institute of Technology Kanpur

2025-04-27 0 0 530.01KB 14 页 10玖币

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Conformal Fisher information metric with torsion

Kunal Pal,1, ∗Kuntal Pal,1, †and Tapobrata Sarkar1, ‡

1Department of Physics, Indian Institute of Technology Kanpur,

Kanpur 208016, India

We consider torsion in parameter manifolds that arises via conformal transformations of

the Fisher information metric, and deﬁne it for information geometry of a wide class of

physical systems. The torsion can be used to diﬀerentiate between probability distribution

functions that otherwise have the same scalar curvature and hence deﬁne similar geometries.

In the context of thermodynamic geometry, our construction gives rise to a new scalar - the

torsion scalar deﬁned on the manifold, while retaining known physical features related to

other scalar quantities. We analyse this in the context of the Van der Waals and the Curie-

Weiss models. In both cases, the torsion scalar has non trivial behaviour on the spinodal

curve.

∗kunalpal@iitk.ac.in

†kuntal@iitk.ac.in

‡tapo@iitk.ac.in

arXiv:2210.04759v1 [physics.class-ph] 10 Oct 2022

I. INTRODUCTION AND MOTIVATION

The Fisher information metric (FIM) corresponding to a probability distribution function (PDF) is ubiquitous

in the study of information geometry [1]. Variants have been studied in physics for decades, in systems ranging

from ﬂuids, spin chains and black holes. The literature on the topic is by now vast, and some representative works

can be found in [2]-[7] and references therein. For reviews, see [8],[9].

Recall that a PDF Pξ(x) is considered to be parameterised by real parameters ξ1,··· , ξn∈ <n, and is a function

of stochastic (or physical) variables x(x∈χ, the space of the physical variables). It is assumed that the PDF is

well behaved, i.e., it is a C∞function, and satisﬁes

Pξ(x)≥0∀x∈χ , ˆPξ(x)dx = 1 .(1)

Then the Fisher information metric corresponding to this PDF is deﬁned by

gij =Eξ∂ilξ∂jlξ ,(2)

where lξ(x) = l(x;ξ) = ln Pξ(x) and Eξrefers to the expectation value of ξwith weight Pξ(x). Also, partial

derivatives are with respect to ξ, so that the FIM is deﬁned on an n×nmanifold.

Given a PDF, it is in general straightforward to deﬁne the FIM (we can write it in a closed form provided that

the integrals are analytically tractable), but the inverse processes, i.e., ﬁnding a PDF from a given FIM is not

unique [10]. The diﬃculty lies in the fact that there can be two (in principle inﬁnitely many) diﬀerent PDFs that

give same FIM. Moreover, the FIM may have extra symmetries that were not originally present in the PDF [11]. A

well known illustration is provided by the Gaussian and the Cauchy distributions,

PG(x;σ, µ) = 1

2π√σexp −(x−µ)2

2σ2, PC(x;x0, γ) = 1

πγ

γ2+ (x−x0)2,(3)

respectively, where in the deﬁnition of PG,σand µdenote respectively the standard deviation and the mean. The

FIMs of the two metrics are similar :

ds2

G=dµ2+ 2dσ2

σ2, ds2

C=dx2

0+dγ2

2γ2,(4)

and both are Euclidean versions of the AdS2metric, i.e., are hyperbolic spaces with constant curvatures. Clearly

then, the map from space of PDFs to FIM is many to one, in principle, inﬁnity to one. Recently, this problem has

been addressed in [10], [11]. One of the purposes of this paper is to oﬀer a simple resolution by considering metrics

related to ds2

Gand ds2

Cvia conformal transformations (CTs) and further by introducing torsion on the parameter

manifold. We will then discuss the implications of these in ﬂuid and spin systems.

In this context, note that the exponential class of PDFs mentioned above, can in general be written as

P(x, θ) = exp hC(x) + θiFi(x)−Ψ(θ)i,(5)

where C(x), Fi(x) are functions of the physical variable xonly, and Ψ(θ) is a function of θonly. Here θiare called

the canonical coordinates for this distribution, and Ψ(θ) is called the potential, which is also the normalisation

factor of the PDF. For example, for the Gaussian distribution, we can identify the potential function to be

ΨG(µ, σ) = µ2

2σ2+ ln(√2πσ).(6)

We are interested in the exponential distribution for two main reasons : ﬁrstly, the calculation of the FIM in this

case becomes simple, since it depends only on the second derivatives of the potential function, taken with respect

to the canonical coordinates, i.e., gij =∂i∂jΨ(θ). Secondly, if we consider a physical system in its thermodynamic

limit (where the particle number N→ ∞) then the PDF can be thought to satisfy a Boltzmann distribution at

inverse temperature β=1

T, namely, P(x, θ) = Z−1e−βH(x,θ), where H(x, θ) is the Hamiltonian of the system, and

Zis the partition function. This PDF thus belongs to the exponential family : P(x, θ) = exp[βH(x, θ)−ln Z(θ)].

Here θirepresent coordinates of the parameter space, which can be coupling constants of the theory, or can be the

thermodynamic variables. Since the partition function of a thermodynamic system is ln Z=−βF , with Fbeing

the free energy, then deﬁning the reduced free energy per site f=−βF/N, one obtains the metric on the parameter

manifold to be gij =∂i∂jf(θ).

In the context of thermodynamic systems, scalar quantities such as the Ricci scalar Rand the expansion scalar

Θ of a geodesic congruence constructed out of the information metric play a crucial role. Several properties of such

scalars are well known in two dimensional systems, for example,

A. Generally, Rdiverges along the spinodal curve.

B. The sign of Rcaptures the nature of interactions in a system (attractive or repulsive).

C. The universal scaling relation R∼λ−2near criticality, where λis an aﬃne parameter measuring geodesic length.

D. The universal scaling relation Θ ∼λ−1near criticality.

E. The relation R∼ldnear criticality, where lis the correlation length.

F. The equality of Rat ﬁrst order phase transitions (which is an alternative characterisation of such transitions and

distinct from, say, the Maxwell construction in Van der Waals ﬂuids).

One can reasonably demand that any modiﬁcation to the information metric should keep the features Ato D

intact, as these follow purely from geometry. For example, a CT that depends on the PDF would produce diﬀerent

(distinguishable) metrics starting from a possibly degenerate set, but simply doing this would cause undesirable

changes in R(see eq. (8) below) that might, for example, violate these features. A similar situation occurs with Θ,

as we explain via eq. (10). Importantly, for relations Eand Fto be physically justiﬁable, Rshould have dimensions

of volume, due to which thermodynamic geometry is always deﬁned via potentials per unit volume (for a more

detailed discussion in the context of black holes, see [12],[13]).

Keeping these issues in mind, in this paper we shall introduce a new metric on the parameter manifold which is

conformally related to the known FIM, but in which Ris invariant under the CT and so is Θ, so that properties A

to Dare retained. As explained below, for this to be true, we have to drop the assumption of symmetric connection

on the parameter manifold i.e., we have to consider torsion as well as curvature on the manifold. If we assume a

particular form of transformation of the connection and torsion components under a CT, then it can be shown that

the curvature tensor and the curvature scalar are related by a benign algebraic factor (that does not contribute to

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ConformalFisherinformationmetricwithtorsionKunalPal,1,KuntalPal,1,yandTapobrataSarkar1,z1DepartmentofPhysics,IndianInstituteofTechnologyKanpur,Kanpur208016,IndiaWeconsidertorsioninparametermanifoldsthatarisesviaconformaltransformationsoftheFisherinformationmetric,anddeneitforinformationgeometryofa...

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Conformal Fisher information metric with torsion Kunal Pal1Kuntal Pal1yand Tapobrata Sarkar1z 1Department of Physics Indian Institute of Technology Kanpur

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