geometric structures in statistical physics · 2017. 7. 17. · they are fundamental in the...

IntroductionGeometric micro-macro mappings

Tropical limit in statistical physics

Geometric structures in statistical physics:

surfaces, amoebas and tropical limit

Mario Angelelli

Department of Mathematics and Physics “E. De Giorgi”, University of Salento & INFN, 73100 Lecce (IT)

Ph.D. course in Physics and Nanosciences - XXIX cycle

Advisor: Prof. B. G. KonopelchenkoCo-Advisor: Prof. G. Landolfi

Coordinator of the Ph. D. Course: Prof. R. Rinaldi

Mario Angelelli Geometric structures in statistical physics

Introduction: statement of the problem

Partition function and free energy are fundamental concepts in physics:

They encode the physical information of complex systems, balancing statistical datawith physical requirements (e.g. observed values of macroscopic quantities).

They allow to derive physical characteristics (e.g., generating correlation functions).

They are fundamental in the description of several ensembles in statistical physics.

They have a large number of extensions: discrete (usual sum), countable (series),continuous (integral), generalizations to quantum settings (trace operator algebras)and field theories (functional integration).

Now they are applied in many branches of science, e.g. mathematics, biology,information theory, economics, AI, . . .

!!! Cases of explicit evaluation of partition function are very rare, due to severalcomplications: (exponentially) hard computations, occurrence of collective phenomena,dependence from boundary conditions, sensitivity to initial conditions for dynamicalsystems, random behaviour etc.

I Our approach: extract distinctive features from partition function and free energy,figure out main characteristics using geometry, investigate composition using algebra.

Goals of the Ph.D. project

Extract relevant features of P.F. and F.E. for a large class of systems, in order notto rely on particular features of specific models;

provide simple tools to explore and characterize relevant physical properties (e.g.,role of interactions, symmetries);

identify phenomena more suitable for this description;

connect di↵erent techniques and contexts that share similarities with thisdescription; applications beyond statistical physics (e.g., integrable systems).

For more details on activities during the Ph. D. course, see ?

Geometry and statistical mappings

Geometric descriptions of statistical objects: statistical hypersurfaces and amoebas.

Tropical algebra: definitions, applications to statistical physics. How to extractunderlying tropical structures in statistical models and formalize them.

Motivations

How to identify and, then,relate the microscopic andmacroscopic sectors?

How to describe more generalstatistical “models”beyond thecanonical ensemble?

Once interactions areincluded, how can they bestudied?

Desiderata

So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.

Perspective

Focus on relations between spaces and see how theirinformation contents are related.

Motivations

Desiderata

So we need to take more spaces and a mappingbetween them

where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.

Perspective

Motivations

Desiderata

So we need to take more spaces and a mappingbetween them

where the microscopic sector dependson a set of parameters and with extra-structures,e.g. metric, which measure deviation from an idealmodel.

Perspective

Motivations

Desiderata

So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters

and with extra-structures,e.g. metric, which measure deviation from an idealmodel.

Perspective

Motivations

Desiderata

So we need to take more spaces and a mappingbetween them where the microscopic sector dependson a set of parameters

and with extra-structures,e.g. metric, which measure deviation from an idealmodel.

Perspective

Motivations

Desiderata

Perspective

Motivations

Desiderata

Perspective

Free-energy-oriented approach: basic statistical mappings

We realize the relation between spaces as a mapping relating their information contents.

Physical insight from partition function Z and free energy F

F = �kB

T · lnZ = �kB

T ln

0

@X

{n}

e�

E{n}k

B

T

1

A . (1)

I Extend free energy properties to a more general class of statistical mappingsintroducting f↵(x)

F (x1, . . . , xn) = ln

NX

↵=1

e f↵(x1,...,xn)!. (2)

Geometry

(2) can be seen as a n-dimensional hypersurface (statistical hypersurface) embedded ina (n + 1)-dimensional ambient space. If the ambient space is endowed with a metricstructure, this hypersurface inherits metric properties.

We realize the relation between spaces as a mapping relating their information contents.Physical insight from partition function Z and free energy F

F = �kB

T · lnZ = �kB

T ln

0

@X

{n}

e�

E{n}k

B

T

1

A . (1)

F (x1, . . . , xn) = ln

NX

↵=1

e f↵(x1,...,xn)!. (2)

Geometry

F = �kB

T · lnZ = �kB

T ln

0

@X

{n}

e�

E{n}k

B

T

1

A . (1)

F (x1, . . . , xn) = ln

NX

↵=1

e f↵(x1,...,xn)!. (2)

Geometry

F = �kB

T · lnZ = �kB

T ln

0

@X

{n}

e�

E{n}k

B

T

1

A . (1)

F (x1, . . . , xn) = ln

NX

↵=1

e f↵(x1,...,xn)!. (2)

Geometry

Main advantages

Free energy-focused approach: it gives a clear picture of its geometriccharacteristics for a wide class of models.

Alternative approach w.r.t. information geometry (or other statistical manifolds),that is oriented towards statistical inference ?

. . . so it is more suited to relevant parametrizations of physical systems (e.g.super-ideal mappings are non-degenerate).

It identifies characterizing objects: Gauss-Kronecker curvature.

Figure: Plane and cylinder are both flat, but the ways they are embedded in R3 are di↵erent: they have di↵erent shapes. TheGauss-Kronecker (G-K) curvature (determinant of the shape operator) is an intrinsic description of immersed hypersurfaces (upto (�1)n).

Main advantages

Basic statistical mappings: main results

For statistical hypersurfaces, geometric characteristics are linked to probability andassociated moments: they depend on averages Q :=

P↵ w↵Q↵ and correlations

Q,P :=P

↵ w↵Q↵P↵ w.r.t. Gibbs distribution w↵ := ef↵(x)�F (x).

Main results: linear f↵(x) = b↵ +nX

i=1

a↵ixi

KG�K = 0 i↵ there exists ~x0 2 Rn : (a · ~x0)` is independent of ` = 1, . . . ,N. If the

G-K curvature vanishes, then there exists a Killing vector field K :=nX

i=1

ci

@@x

i

,

c1, . . . , cn constants.

For a super-ideal statistical hypersurface xn+1 = ln

nX

i=1

exi

!G-K curvature

vanishes, while mean and scalar curvatures take values in

0 ⌦ n � 1pn(n + 1)

, 0 R (n � 1)(n � 2)n(n + 1)

. (3)

Q,P :=P

i=1

a↵ixi

i=1

ci

@@x

i

,

nX

i=1

exi

!G-K curvature

0 ⌦ n � 1pn(n + 1)

, 0 R (n � 1)(n � 2)n(n + 1)

. (3)

Q,P :=P

i=1

a↵ixi

i=1

ci

@@x

i

,

nX

i=1

exi

!G-K curvature

0 ⌦ n � 1pn(n + 1)

, 0 R (n � 1)(n � 2)n(n + 1)

. (3)

Main results on statistical hypersurfaces

Main physical ideal models have vanishing of G-K curvature. The converse is not true:also non-linear f↵ can give vanishing G-K. It is related to the invariance w.r.t. choice ofthe“ground energy”. If it holds also for independent variables x 7! x+ C thenK

G�K = 0.

Main results: non-linear f↵

General formulas are provided for metric objects ? .

Entropy is expressed as a coupling between the hypersurface and its normal bundle.For homogeneous interactions of degree d

S =p

det g�!X ·�!N + (d � 1)f̄ (4)

Metric on the hyper-surface

Position vector(x1, . . . , xn,F )

Unit normal vectorwith a given orienta-tion

mean value f̄ =PN

↵=1 w↵f↵

Metric characterization of s.-i. mapping: under the rank and the closure hypotheses

g̃ij

(x, t) = �ij

+ hi

(x, t) · hj

(x, t),@h

i

@xj

= C · (�ij

hj

� hi

hj

) (5)

(t deformation parameter) there exists a potential function F̃ such that hi

= @x

i

F̃and F̃ (x, t) = log(�(t) +

Pn

i=1 ex

i

�↵i

(t)).

G�K = 0.Main results: non-linear f↵

S =p

det g�!X ·�!N + (d � 1)f̄ (4)

mean value f̄ =PN

↵=1 w↵f↵

g̃ij

(x, t) = �ij

+ hi

(x, t) · hj

(x, t),@h

i

@xj

= C · (�ij

hj

� hi

hj

) (5)

= @x

i

F̃and F̃ (x, t) = log(�(t) +

Pn

i=1 ex

i

�↵i

(t)).

S =p

det g�!X ·�!N + (d � 1)f̄ (4)

mean value f̄ =PN

↵=1 w↵f↵

g̃ij

(x, t) = �ij

+ hi

(x, t) · hj

(x, t),@h

i

@xj

= C · (�ij

hj

� hi

hj

) (5)

= @x

i

F̃and F̃ (x, t) = log(�(t) +

Pn

i=1 ex

i

�↵i

(t)).

S =p

det g�!X ·�!N + (d � 1)f̄ (4)

mean value f̄ =PN

↵=1 w↵f↵

g̃ij

(x, t) = �ij

+ hi

(x, t) · hj

(x, t),@h

i

@xj

= C · (�ij

hj

� hi

hj

) (5)

= @x

i

F̃and F̃ (x, t) = log(�(t) +

Pn

i=1 ex

i

�↵i

(t)).

S =p

det g�!X ·�!N + (d � 1)f̄ (4)

mean value f̄ =PN

↵=1 w↵f↵

g̃ij

(x, t) = �ij

+ hi

(x, t) · hj

(x, t),@h

i

@xj

= C · (�ij

hj

� hi

hj

) (5)

= @x

i

F̃and F̃ (x, t) = log(�(t) +

Pn

i=1 ex

i

�↵i

(t)).

Some results on statistical hypersurfaces

In case of perturbations of P ideal subsystems due to "⌧ 1, Gauss-Kroneckercurvature still vanishes at first order in ". Higher orders are in generalnon-vanishing.

Several examples of non-linear interactions are provided, both with vanishing G-Kcurvature (cilindrical-type symmetry) and not.

Phase singularities are described by singularities of the metric. They can behidden or visible from the curvature p.o.v., depending on the specific form of f↵s.There can be first-order phase transitions (singularity of the metric) and higherorder (singularity of the curvature). Characterizing cases are explored.

For more details, see also [A. & Konopelchenko, 2016]

Statistical amoebas: motivations

More subsystems:

N1X

↵=1

eg↵(x) = e fN1+N2+1(x) =N2X

�=1

eh� (x). (6)

Relaxed (physical) assumption on g↵: negative degeneracies of energy levels,negative probabilities [Wigner, 1932; Dirac, 1942; Feynman, 1987; Blizard,1990; Burgin, 2010]).

Singularity of free energy at the zeros of the partition function , phase transition[Lee & Yang, 1952; Lee & Yang, 1952, b; Newman & Schulman, 1980].

+Real non-positive definite partition functions combine reality constraints (typical ofthermodynamics) with the investigation of metastable states in terms of zeros of P.F.

More subsystems:

N1X

↵=1

eg↵(x) =

e fN1+N2+1(x) =

N2X

�=1

eh� (x). (6)

More subsystems:

0 = �N1X

↵=1

e f↵(x) +N2X

�=1

e f� (x). (6)

More subsystems:

0 = �N1X

↵=1

e f↵(x) +N2X

�=1

e f� (x). (6)

More subsystems:

0 = �N1X

↵=1

e f↵(x) +N2X

�=1

e f� (x). (6)

More subsystems:

0 = �N1X

↵=1

e f↵(x) +N2X

�=1

e f� (x). (6)

Statistical amoebas: definitions

If I ✓ {1, . . . ,N}, #I = k and f↵ are real polynomials, then introduce

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

I 7! {1, . . . ,N}\I acts asg↵ 7! �g↵ and Zk(I) 7! �Zk(I).One can fix #I N �#I toavoid such a redundancy.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

I 7! {1, . . . ,N}\I acts asg↵ 7! �g↵ and Zk(I) 7! �Zk(I).One can fix #I N �#I toavoid such a redundancy.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

Hypersurfaces (7) divide Rn indomains. In open subdomains, allfunctions Z

k

(I) have well-defined(positive or negative) sign.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

Hypersurfaces (7) divide Rn indomains. In open subdomains, allfunctions Z

k

(I) have well-defined(positive or negative) sign.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

One can fix an order (e.g.,lexicographic) on k-subsets of [N]and associate with each subdomaina vector

(Sk;�)⌧ := sign

Z(I⌧ ; x) : 1 ⌧

Nk

!!

(8)evaluated at an interior point x.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

One can fix an order (e.g.,lexicographic) on k-subsets of [N]and associate with each subdomaina vector

(Sk;�)⌧ := sign

Z(I⌧ ; x) : 1 ⌧

Nk

!!

(8)evaluated at an interior point x.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

Stability domain Dk+ ✓ Rn:

Zk

(I) > 0 for all I with #I = k.Instability domain D

k�: maximalnumber of negative Z

k

(I).Hypersurfaces of the k-stratum areconfined in the zeros confinementdomain

ZCDk

:= Rn\(Dk+ [Dk�).

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

Stability domain Dk+ ✓ Rn:

Zk

(I) > 0 for all I with #I = k.Instability domain D

k�: maximalnumber of negative Z

k

(I).Hypersurfaces of the k-stratum areconfined in the zeros confinementdomain

ZCDk

:= Rn\(Dk+ [Dk�).

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

The set Dk+ [ ZCDk is the

statistical k-amoeba and itsboundary contains its extremalpoints.

Z(I; x) := �X

↵2I

e f↵(x) +X

�/2I

e f� (x) = 0. (7)

The set Dk+ [ ZCDk is the

statistical k-amoeba and itsboundary contains its extremalpoints.

Statistical amoebas: main results for polymonial f↵

Theorem

The maximum number of �1 in Sk

at fixed k <N2

is equal to�N�1k�1�. If 2k = N then

the number of �1 signs in SN

2is identically equal to

�2k�1

k

�on Rn\Z

sing, N2. Then, the

integral characteristic

S̄

k;� =1�N

k

� ·(Nk

)X

⌧=1

(Sk;�)⌧ (8)

takes values in the interval⇥1� 2 k

N

; 1⇤. The maximum 1 (resp. minimum �1 + 2 k

N

) isreached in the stability (resp. instability) domain D

k+ (resp. Dk�).

Theorem

The maximum number of �1 in Sk

at fixed k <N2

is equal to�N�1k�1�. If 2k = N then

the number of �1 signs in SN

2is identically equal to

�2k�1

k

�on Rn\Z

sing, N2. Then, the

integral characteristic

S̄

k;� =1�N

k

� ·(Nk

)X

⌧=1

(Sk;�)⌧ (8)

takes values in the interval⇥1� 2 k

N

; 1⇤. The maximum 1 (resp. minimum �1 + 2 k

N

) isreached in the stability (resp. instability) domain D

k+ (resp. Dk�).

Proposition

If 1 k < k̂ �N2

⌫, then

Dk̂+ ✓ Dk+, Dk� ✓ Dk̂�. (9)

Proposition

If 1 k < k̂ �N2

⌫, then

Dk̂+ ✓ Dk+, Dk� ✓ Dk̂�. (8)

Statistical amoebas: main results

Lemma

Two hypersurfaces (7) belonging to the same stratum Zsing,k and di↵erent I1 and I2

intersect at finite x only if I1 \ I2 6= ;.

Proposition

If {f↵(x) : ↵ 2 [N]} are pairwise di↵erent polynomial functions, then Zsing,k̂ lies inside a

region of Rn delimited by some components of Zsing,k , 1 k < k̂

�N2

⌫.

Proposition

The equilibrium domain Dk+ is the set of all points x such that, for all

N � k + 1 g N, there exists a non-degenerate polygon with g sides of lengthsX↵2I1

e f↵(x), . . . ,X

↵2Ig

e f↵(x) where {I1, . . . , Ig} is any partition of [N] in non-empty

subsets.

Proposition

For homogeneous functions, connected components of Dk� are unbounded for all

1 k N2

� 1.

Spin interpretation: “microcanonical”probability wN,k =

�N

k

��1on each domain,

“canonical”probability with energy (8).

Connections with algebraic geometry: s.a. generalize (non-)lopsided sets, whichare good approximations of an algebraic amoeba. The introduction of amoebastakes us to the realm of tropical mathematics.

For more details, see also [A. & Konopelchenko, 2016, b]

Proposition

The equilibrium domain Dk+ is the set of all points x such that, for all

N � k + 1 g N, there exists a non-degenerate polygon with g sides of lengthsX↵2I1

e f↵(x), . . . ,X

↵2Ig

e f↵(x) where {I1, . . . , Ig} is any partition of [N] in non-empty

subsets.

Proposition

For homogeneous functions, connected components of Dk� are unbounded for all

1 k N2

� 1.

Spin interpretation: “microcanonical”probability wN,k =

�N

k

��1on each domain,

“canonical”probability with energy (8).

Connections with algebraic geometry: s.a. generalize (non-)lopsided sets, whichare good approximations of an algebraic amoeba. The introduction of amoebastakes us to the realm of tropical mathematics.

For more details, see also [A. & Konopelchenko, 2016, b]

Tropical limit: introduction

Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .

Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction

(x, t) = A(x, t) · exp✓iS(x, t)

~

◆

at ~ ! 0.

(9)

Semiclassical limit of awavefunction

Linear di↵erential equations generally becomes non-linear in this limit (e.g.Schrödinger Hamilton-Jacobi) and the superposition principle is lost unless algebraicoperations change too!

z =NX

`=1

x` eZ"/" =

NX

`=1

eX`,"/". (10)Family of “additions”

At "! 0+: semiring ?(11)

Tropical limit: transformation which results from a limit procedure with an essentialsingularity ? .Physical rationale: study of objects for small values of characterizing certain“constants”(! deformations). Ex: semiclassical limit of a wavefunction

(x, t) = A(x, t) · exp✓iS(x, t)

~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

(x, t) =

A(x, t) ·

exp

✓

i

S(x, t)~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

(x, t) =

A(x, t) ·

exp

✓

i

S(x, t)~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

(x, t) =

A(x, t) ·

exp

✓

i

S(x, t)~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

At "! 0+: semiring ?

Xtrop

� Ytrop

:= lim"!0

✓" log

✓exp

Xtrop

"· exp Ytrop

"

◆◆= X

trop

+ Ytrop

, (11)

(x, t) =

A(x, t) ·

exp

✓

i

S(x, t)~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

At "! 0+: semiring ?

Xtrop

� Ytrop

:= lim"!0

✓" log

✓exp

Xtrop

"+ exp

Ytrop

"

◆◆= max{X

trop

,Ytrop

}. (11)

(x, t) =

A(x, t) ·

exp

✓

i

S(x, t)~

◆at ~ ! 0. (9)

z =NX

`=1

x` eZ"/" =

NX

`=1

At "! 0+: semiring with idempotent addition: a� a = max{a, a} = a ?

Xtrop

� Ytrop

:= lim"!0

✓" log

✓exp

Xtrop

"+ exp

Ytrop

"

◆◆= max{X

trop

,Ytrop

}. (11)

Our starting point is

kB

! 0. (12)

In more details,

Definition

Given a physical statistical system specified by a partition function and an associatedfree energy, its tropical limit is defined as the simultaneous limit for k

B

and Avogadronumber N

A

such that their product is kept constant, i.e.

kB

! 0+, NA

! 1 : kB

· NA

=: R is constant.

F = �kB

T lnX

n

gn

exp

✓� EnkB

T

◆�! F

trop

(T ) = �TM

n

✓�Fn

T

◆

g

n

=: exp

S

n

k

B

!,

F

n

:= En

� T · Sn

%

kB

! 0. (12)

In more details,

Definition

B

A

kB

! 0+, NA

! 1 : kB

· NA

=: R is constant.

F = �kB

T lnX

n

gn

exp

✓� EnkB

T

◆�! F

trop

(T ) = �TM

n

✓�Fn

T

◆

g

n

=: exp

S

n

k

B

!,

F

n

:= En

� T · Sn

%

kB

! 0. (12)

In more details,

Definition

B

A

kB

! 0+, NA

! 1 : kB

· NA

=: R is constant.

F = �kB

T lnX

n

gn

exp

✓� EnkB

T

◆�! F

trop

(T ) = �TM

n

✓�Fn

T

◆

g

n

=: exp

S

n

k

B

!,

F

n

:= En

� T · Sn

%

kB

! 0. (12)

In more details,

Definition

B

A

kB

! 0+, NA

! 1 : kB

· NA

=: R is constant.

F = �kB

T lnX

n

gn

exp

✓� EnkB

T

◆�! F

trop

(T ) = �TM

n

✓�Fn

T

◆

g

n

=: exp

S

n

k

B

!,

F

n

:= En

� T · Sn

%

Motivations

On the basis of (2), particular physical phenomena can be studied in a new perspective[A. & Konopelchenko, 2015].

tropical dictionary for thermodynamic and statistical quantities, e.g. tropical freeenergy and tropical Gibbs probability distribution. It provides a simplecombinatorial framework to investigate the system.

Non-trivial phenomena come out in case of exponential degenerations gn

ofenergy levels through the reparametrization

gn

:= exp

✓Sn

kB

◆. (13)

Many phenomena related to exponential degenerations: residual entropy [Pauling,1935], topological order [Wen, 2013], frustrated systems ( [Diep, 2005; Lieb & Wu,1972], mainly spin ices [Gingras, 2011] and spin glasses [Parisi et al., 1987]), HEP(QCD [Blanchard et al., 2004]), cosmology [Carlip, 1997]. Some of these (residualentropy, limiting temperature, ultrametricity) are discussed in the tropical algebraformalism.

Motivations

gn

:= exp

✓Sn

kB

◆. (13)

Motivations

gn

:= exp

✓Sn

kB

◆. (13)

Motivations

gn

:= exp

✓Sn

kB

◆. (13)

Motivations

These features are unseen in previous approach to tropical limit as a low temperaturelimit [Marcolli & Thorngren, 2012; Kapranov, 2011].

conceptual di↵erence: T ! 0 describes a boundary of the phase space, kB

! 0 is adi↵erent statistics which filters some aspects to highlight other ones;

preserves thermodynamic relations leaving T as a free parameter;

freedom in varying temperature can be used to study previous phenomena in greatgenerality including negative and limiting temperatures [Rumer, 1960; Atick &Witten, 1988]. ?

Motivations

Some results: thermodynamics

Etrop

:= limk

B

!0E , F

trop

:= limk

B

!0F and w

n,trop = limk

B

!0(k

B

· lnwn

) . . .

,

Ftrop

(T ) = �TM

n

✓�Fn

T

◆(14)

(= min{F1,F2, ..,Fn, ..} if T > 0).Gibbs’ distribution:

wn

=

exp

✓� EnkT

◆

X

m

gm

exp

✓�EmkT

◆ wn,trop = �Sn + Ftrop � Fn

T. (15)

Tropical Gibbs’ distribution for levels is

Wn,trop = wn,trop + Sn =

Ftrop

� Fn

T(16)

which satisfies the normalization conditionM

n

Wn,trop = 0. (17)

Etrop

:= limk

B

!0E , F

trop

:= limk

B

!0F and w

n,trop = limk

B

!0(k

B

· lnwn

) . . . ,

Ftrop

(T ) = �TM

n

✓�Fn

T

◆(14)

wn

=

exp

✓� EnkT

◆

X

m

gm

exp

✓�EmkT

T. (15)

Ftrop

� Fn

T(16)

n

Wn,trop = 0. (17)

Etrop

:= limk

B

!0E , F

trop

:= limk

B

!0F and w

n,trop = limk

B

!0(k

B

· lnwn

) . . . ,

Ftrop

(T ) = �TM

n

✓�Fn

T

◆(14)

wn

=

exp

✓� EnkT

◆

X

m

gm

exp

✓�EmkT

T. (15)

Ftrop

� Fn

T(16)

n

Wn,trop = 0. (17)

Etrop

:= limk

B

!0E , F

trop

:= limk

B

!0F and w

n,trop = limk

B

!0(k

B

· lnwn

) . . . ,

Ftrop

(T ) = �TM

n

✓�Fn

T

◆(14)

wn

=

exp

✓� EnkT

◆

X

m

gm

exp

✓�EmkT

T. (15)

Ftrop

� Fn

T(16)

n

Wn,trop = 0. (17)

Singular locus

Singular locus: coincidence of at least two dominant terms Fi

= Fj

� Fk

,i 6= j 6= k 6= i , arising for certain values of intensive parameter. One has singular points(i.e. transition temperatures)

T ⇤ =En0 � Em0

Sn0 � Sm0

(18)

which are related by

(Si

� Sk

)T ⇤ik

+ (Sk

� Sl

)T ⇤kl

+ (Sl

� Si

)T ⇤li

= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)Near the singular temperature T ⇤ it is �W

trop

⇠ (T ⇤ � T )

�Wtrop

= Wn0,trop(T )�Wm0,trop(T ) = Fm0 � Fn0T = �Strop ·

✓T ⇤

T� 1◆

(20)

and non-dominant terms become relevant at T ⇤: expanding

lnwn

=w

n,trop

kB

+ ln w̃n

+ ..., (21)

#{n : Fn

= Ftrop

} = m , lnwn

= � Snk

B

� Fn(T⇤)�Fn0 (T⇤)kT

⇤ � lnm +O(kB).

Singular locus

= Fj

� Fk

T ⇤ =En0 � Em0

Sn0 � Sm0

(18)

(Si

� Sk

)T ⇤ik

+ (Sk

� Sl

)T ⇤kl

+ (Sl

� Si

)T ⇤li

= 0, i 6= k 6= l 6= i , i , k, l = 1, ..., n. (19)

Near the singular temperature T ⇤ it is �Wtrop

⇠ (T ⇤ � T )

�Wtrop

✓T ⇤

T� 1◆

(20)

lnwn

=w

n,trop

kB

+ ln w̃n

+ ..., (21)

#{n : Fn

= Ftrop

} = m , lnwn

= � Snk

B

⇤ � lnm +O(kB).

Singular locus

= Fj

� Fk

T ⇤ =En0 � Em0

Sn0 � Sm0

(18)

(Si

� Sk

)T ⇤ik

+ (Sk

� Sl

)T ⇤kl

+ (Sl

� Si

)T ⇤li

trop

⇠ (T ⇤ � T )

�Wtrop

✓T ⇤

T� 1◆

(20)

lnwn

=w

n,trop

kB

+ ln w̃n

+ ..., (21)

#{n : Fn

= Ftrop

} = m , lnwn

= � Snk

B

⇤ � lnm +O(kB).

Singular locus

= Fj

� Fk

T ⇤ =En0 � Em0

Sn0 � Sm0

(18)

(Si

� Sk

)T ⇤ik

+ (Sk

� Sl

)T ⇤kl

+ (Sl

� Si

)T ⇤li

trop

⇠ (T ⇤ � T )

�Wtrop

✓T ⇤

T� 1◆

(20)

lnwn

=w

n,trop

kB

+ ln w̃n

+ ..., (21)

#{n : Fn

= Ftrop

} = m , lnwn

= � Snk

B

⇤ � lnm +O(kB).

Nested tropical limit

The function e� 1

k

B is not analytic around kB

= 0, thus standard perturbative tools (e.g.,series expansion) may fail.

Proposition

Zeroth and first order corrections of F = F (kB

) correspond to data Ftrop

and m as in(21), respectively. The `-th order coe�cient in the Taylor expansion of F near k

B

= 0vanishes, for all ` � 2.However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?

Z(x) =eµ0(x) ·⇣⌫0 + e

µ1(x)�µ0(x) ·⇣⌫1 + e

µ2(x)�µ1(x) · (⌫2 + . . .· · · ·

⇣⌫L�2 + e

µL�1(x)�µL�2(x)·

⇣vL�1 + e

µL

(x)�µL�1(x) · ⌫

L

⌘. . .⌘⌘⌘⌘

. (22)

The function e� 1

k

Proposition

B

= 0vanishes, for all ` � 2.

However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?

Z(x) =eµ0(x) ·⇣⌫0 + e

µ1(x)�µ0(x) ·⇣⌫1 + e

µ2(x)�µ1(x) · (⌫2 + . . .· · · ·

⇣⌫L�2 + e

µL�1(x)�µL�2(x)·

⇣vL�1 + e

µL

(x)�µL�1(x) · ⌫

L

⌘. . .⌘⌘⌘⌘

. (22)

The function e� 1

k

Proposition

B

= 0vanishes, for all ` � 2.However, one can use perturbation theory and additional structures, e.g. the nestingform of the P.F. to apply the tropical limit at di↵erent levels ?

Z(x) =eµ0(x) ·⇣⌫0 + e

µ1(x)�µ0(x) ·⇣⌫1 + e

µ2(x)�µ1(x) · (⌫2 + . . .· · · ·

⇣⌫L�2 + e

µL�1(x)�µL�2(x)·

⇣vL�1 + e

µL

(x)�µL�1(x) · ⌫

L

⌘. . .⌘⌘⌘⌘

. (22)

Some figures

Ex: 3-level case: classifying cases.

(a) Ftrop

in three level case,S1 < S2 < S3,T

⇤23 > T

⇤13 > T

⇤12.

(b) Ftrop

⇤23 < T

⇤13 < T

⇤12.

(c) Ftrop

in three level case,S3 < S1 < S2: at0 < T < T⇤12, Etrop = E1,S

trop

= S1.

What if the spectrum is {Ei

: i 2 N} or {Ei

: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist! Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?

Some figures

(a) Ftrop

⇤23 > T

⇤13 > T

⇤12.

(b) Ftrop

⇤23 < T

⇤13 < T

⇤12.

(c) Ftrop

trop

= S1.

: i 2 N} or {Ei

: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist!

Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?

Some figures

(a) Ftrop

⇤23 > T

⇤13 > T

⇤12.

(b) Ftrop

⇤23 < T

⇤13 < T

⇤12.

(c) Ftrop

trop

= S1.

: i 2 N} or {Ei

: i 2 Z}? Singularity can occur whenmin{F1,F2, ...} does not exist! Even in Rmax = (R [ {�1},max,+), and hence�1 2 Rmax, Wn,trop does not define a distribution of probabilities obeying the tropicalnormalization condition (17). ?

Tropical limits of statistical hypersurfaces and amoebas

More tropical limits for independent variables (parameters) or dependent ones(“microscopic free energies” f↵).

Dependent variables: xi

= �x?i

for large values of variables xi

� 1, i = 1, . . . , n andx?i

finite.

Statistical hypersurface is piecewise smooth. On the singular locus geometriccharacteristics can be regularized.

For linear f↵:

xn+1 = �x

?n+1,

The metric is non-degenerate in the tropicallimit, the resulting tropical hypersurface ispiecewise-flat.

Nonlinear f↵ require a double scalingtropical limit:

xn+1 = �

d · x?n+1.

Highly degenerate (rank 1) metric.

Connected components of the complement of the tropical graph of the first kind(“slow variables”) are unbounded.

= �x?i

� 1, i = 1, . . . , n andx?i

finite.

For linear f↵:

xn+1 = �x

?n+1,

xn+1 = �

d · x?n+1.

= �x?i

� 1, i = 1, . . . , n andx?i

finite.

For linear f↵:

xn+1 = �x

?n+1,

xn+1 = �

d · x?n+1.

= �x?i

� 1, i = 1, . . . , n andx?i

finite.

For linear f↵:

xn+1 = �x

?n+1,

xn+1 = �

d · x?n+1.

= �x?i

� 1, i = 1, . . . , n andx?i

finite.

For linear f↵:

xn+1 = �x

?n+1,

xn+1 = �

d · x?n+1.

d > 1 is the highest degreeamong the homogeneous partsof the interactions {f↵}.

= �x?i

� 1, i = 1, . . . , n andx?i

finite.

For linear f↵:

xn+1 = �x

?n+1,

xn+1 = �

d · x?n+1.

T

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