text: 2 the traditional combustion model of a detonation front leading shock fast reaction consisting of an inﬁnitely thin leading shock and a zone induction of energy release of ﬁnite thickness.20 Such a model has fuel burnt (slow been originally developed for gaseous detonations, which mixture gas D are quite strong and propagate with speed exceeding the reaction) sound speed in gases by an order of magnitude. However, it is not rigorous applying such a model to magnetic det- onation, which is extremely weak by combustion scales P T (although quite strong when compared to magnetic deﬂa- D gration). In contrast to strong shocks in gases, which are inﬁnitely thin from the hydrodynamic point of view, a weak shock exhibits a continuous structure controlled by transport processes such as thermal conduction and/or viscosity.21 The same property should be naturally ex- z pected for magnetic detonation. The purpose of the present work is to provide accurate FIG. 2: Schematic of strong gaseous detonation with char- description of magnetic detonation structure in crystals acteristic proﬁles of temperature and pressure. of nanomagnets by taking into account thermal conduc- tion and volume viscosity. Here we show that, in con- trast to the previously suggested model of Ref. [2], the certain (induction) time for self-acceleration of chemical transport processes result in smooth proﬁles of the most reactions in a gas parcel after passing the shock, after important thermodynamical crystal parameters – such as which the chemical reactions become fast and convert temperature, density and pressure – all over the magnetic the fuel mixture into the burning products with release detonation front including the leading weak shock, which of a large amount of energy and strong expansion of the is one of the key regions of magnetic detonation. In the burning gas. case of zero volume viscosity, however, thermal conduc- Here we are interested in a planar stationary one- tion leads to an isothermal discontinuity instead of the dimensional detonation propagating with constant super- shock, for which temperature is continuous while density sonic speed D in a uniform gaseous mixture; we take the and pressure experience jump. detonation front propagating along the z-axis in the neg- The present paper is organized as follows. In Sec. II we ative direction. By adopting the reference frame of the overview basic features of gaseous combustion detonation stationary detonation front, we obtain the fuel mixture needed for proper understanding of magnetic detonation moving with velocity u = D towards the leading shock 0 in crystals of nanomagnets. In Sec. III we present basic of zero thickness; the initial density, pressure and temper- equations describing magnetic detonation, develop the ature are designated by ρ , P and T . The fuel mixture 0 0 0 analytical theory for the most important detonation pa- is compressed in the shock (label ”s”) with strong in- rameters and study magnetic detonation structure con- crease of density, pressure and temperature to ρ > ρ , s 0 trolled by thermal conduction only assuming zero volume P > P , T > T , and with drop of the gas velocity to s 0 s 0 viscosity – such a structure involves isothermal disconti- a new subsonic value u < D. Temperature increase ini- s nuity instead of a shock employed in Ref. [2]. In Sec. IV tiates combustion reactions with release of chemical en- we demonstrate dramatic modiﬁcations of the magnetic ergy, hence leading to even stronger temperature increase detonation structure due to volume viscosity. until it reaches the ﬁnal maximal value in the detonation II. BASIC FEATURES OF GASEOUS products (label ”d”), T > T . At the same time, pres- d s COMBUSTION DETONATION sure and density decrease in the reaction region, ρ < ρ , d s P < P , as illustrated schematically on Fig. 2. d s In this section we remind brieﬂy the most important Modiﬁcations of the gas parameters in a detonation features and the methods of investigation of traditional front are described by the hydrodynamic laws of mass, gaseous combustion detonations, in order to highlight momentum and energy conservation expressed in terms the similarity and diﬀerence between the combustion and of ﬂuxes21 magnetic detonations. By deﬁnition, detonation is a fast supersonic combustion regime, for which preheating of the cold fuel mixture happens due to the leading shock, ρu = const = ρ D, (1) 1 0 see Fig. 2. In turn, shock propagation without decay in combustion detonation is supported by energy release in chemical reactions in the active reaction zone. Typically for combustion, the activation energy of the reactions is quite high, so that reactions develop relatively slow just P + ρu2 = const = P + ρ D2, (2) 2 0 0 after the leading shock in the so-called induction zone (still, much faster than in the fuel mixture). It requires ##LTFigure## ##LTLine## ##LTLine##

text: the

text: traditional

text: combustion

text: model

text: of

text: a

text: detonation

text: front

text: leading

text: shock

text: fast

text: reaction

text: consisting

text: of

text: an

text: inﬁnitely

text: thin

text: leading

text: shock

text: and

text: a

text: zone

text: induction

text: of

text: energy

text: release

text: of

text: ﬁnite

text: Such

text: a

text: model

text: has

text: fuel

text: burnt

text: (slow

text: been

text: originally

text: developed

text: for

text: gaseous

text: detonations,

text: which

text: mixture

text: gas

text: D

text: are

text: quite

text: strong

text: and

text: propagate

text: with

text: speed

text: exceeding

text: the

text: reaction)

text: sound

text: speed

text: in

text: gases

text: by

text: an

text: order

text: of

text: magnitude.

text: However,

text: it

text: is

text: not

text: rigorous

text: applying

text: such

text: a

text: model

text: to

text: magnetic

text: det-

text: onation,

text: which

text: is

text: extremely

text: weak

text: by

text: combustion

text: scales

text: P

text: T

text: (although

text: quite

text: strong

text: when

text: compared

text: to

text: magnetic

text: deﬂa-

text: D

text: gration).

text: In

text: contrast

text: to

text: strong

text: shocks

text: in

text: gases,

text: which

text: are

text: inﬁnitely

text: thin

text: from

text: the

text: hydrodynamic

text: point

text: of

text: view,

text: a

text: weak

text: shock

text: exhibits

text: a

text: continuous

text: structure

text: controlled

text: by

text: transport

text: processes

text: such

text: as

text: thermal

text: conduction

text: and/or

text: The

text: same

text: property

text: should

text: be

text: naturally

text: ex-

text: z

text: pected

text: for

text: magnetic

text: detonation.

text: The

text: purpose

text: of

text: the

text: present

text: work

text: is

text: to

text: provide

text: accurate

text: FIG.

text: Schematic

text: of

text: strong

text: gaseous

text: detonation

text: with

text: char-

text: description

text: of

text: magnetic

text: detonation

text: structure

text: in

text: crystals

text: acteristic

text: proﬁles

text: of

text: temperature

text: and

text: pressure.

text: of

text: nanomagnets

text: by

text: taking

text: into

text: account

text: thermal

text: conduc-

text: tion

text: and

text: volume

text: viscosity.

text: Here

text: we

text: show

text: that,

text: in

text: con-

text: trast

text: to

text: the

text: previously

text: suggested

text: model

text: of

text: Ref.

text: the

text: certain

text: (induction)

text: time

text: for

text: self-acceleration

text: of

text: chemical

text: transport

text: processes

text: result

text: in

text: smooth

text: proﬁles

text: of

text: the

text: most

text: reactions

text: in

text: a

text: gas

text: parcel

text: after

text: passing

text: the

text: shock,

text: after

text: important

text: thermodynamical

text: crystal

text: parameters

text: such

text: as

text: which

text: the

text: chemical

text: reactions

text: become

text: fast

text: and

text: convert

text: temperature,

text: density

text: and

text: pressure

text: all

text: over

text: the

text: magnetic

text: the

text: fuel

text: mixture

text: into

text: the

text: burning

text: products

text: with

text: release

text: detonation

text: front

text: including

text: the

text: leading

text: weak

text: shock,

text: which

text: of

text: a

text: large

text: amount

text: of

text: energy

text: and

text: strong

text: expansion

text: of

text: the

text: is

text: one

text: of

text: the

text: key

text: regions

text: of

text: magnetic

text: detonation.

text: In

text: the

text: burning

text: gas.

text: case

text: of

text: zero

text: volume

text: viscosity,

text: however,

text: thermal

text: conduc-

text: Here

text: we

text: are

text: interested

text: in

text: a

text: planar

text: stationary

text: one-

text: tion

text: leads

text: to

text: an

text: isothermal

text: discontinuity

text: instead

text: of

text: the

text: dimensional

text: detonation

text: propagating

text: with

text: constant

text: super-

text: shock,

text: for

text: which

text: temperature

text: is

text: continuous

text: while

text: density

text: sonic

text: speed

text: D

text: in

text: a

text: uniform

text: gaseous

text: mixture;

text: we

text: take

text: the

text: and

text: pressure

text: experience

text: jump.

text: detonation

text: front

text: propagating

text: along

text: the

text: z-axis

text: in

text: the

text: neg-

text: The

text: present

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: II

text: we

text: ative

text: direction.

text: By

text: adopting

text: the

text: reference

text: frame

text: of

text: the

text: overview

text: basic

text: features

text: of

text: gaseous

text: combustion

text: detonation

text: stationary

text: detonation

text: front,

text: we

text: obtain

text: the

text: fuel

text: mixture

text: needed

text: for

text: proper

text: understanding

text: of

text: magnetic

text: detonation

text: moving

text: with

text: velocity

text: u

text: D

text: towards

text: the

text: leading

text: shock

text: in

text: crystals

text: of

text: nanomagnets.

text: In

text: Sec.

text: III

text: we

text: present

text: basic

text: of

text: zero

text: thickness;

text: the

text: initial

text: density,

text: pressure

text: and

text: temper-

text: equations

text: describing

text: magnetic

text: detonation,

text: develop

text: the

text: ature

text: are

text: designated

text: by

text: ρ

text: P

text: and

text: T

text: The

text: fuel

text: mixture

text: analytical

text: theory

text: for

text: the

text: most

text: important

text: detonation

text: pa-

text: is

text: compressed

text: in

text: the

text: shock

text: (label

text: ”s”)

text: with

text: strong

text: in-

text: rameters

text: and

text: study

text: magnetic

text: detonation

text: structure

text: con-

text: crease

text: of

text: density,

text: pressure

text: and

text: temperature

text: to

text: ρ

text: ρ

text: s

text: trolled

text: by

text: thermal

text: conduction

text: only

text: assuming

text: zero

text: volume

text: P

text: P

text: T

text: T

text: and

text: with

text: drop

text: of

text: the

text: gas

text: velocity

text: to

text: s

text: s

text: viscosity

text: such

text: a

text: structure

text: involves

text: isothermal

text: disconti-

text: a

text: new

text: subsonic

text: value

text: u

text: D.

text: Temperature

text: increase

text: ini-

text: s

text: nuity

text: instead

text: of

text: a

text: shock

text: employed

text: in

text: Ref.

text: In

text: Sec.

text: IV

text: tiates

text: combustion

text: reactions

text: with

text: release

text: of

text: chemical

text: en-

text: we

text: demonstrate

text: dramatic

text: modiﬁcations

text: of

text: the

text: magnetic

text: ergy,

text: hence

text: leading

text: to

text: even

text: stronger

text: temperature

text: increase

text: detonation

text: structure

text: due

text: to

text: volume

text: viscosity.

text: until

text: it

text: reaches

text: the

text: ﬁnal

text: maximal

text: value

text: in

text: the

text: detonation

text: II.

text: BASIC

text: FEATURES

text: OF

text: GASEOUS

text: products

text: (label

text: ”d”),

text: T

text: T

text: At

text: the

text: same

text: time,

text: pres-

text: d

text: s

text: COMBUSTION

text: DETONATION

text: sure

text: and

text: density

text: decrease

text: in

text: the

text: reaction

text: region,

text: ρ

text: ρ

text: d

text: s

text: P

text: P

text: as

text: illustrated

text: schematically

text: on

text: Fig.

text: d

text: s

text: In

text: this

text: section

text: we

text: remind

text: brieﬂy

text: the

text: most

text: important

text: Modiﬁcations

text: of

text: the

text: gas

text: parameters

text: in

text: a

text: detonation

text: features

text: and

text: the

text: methods

text: of

text: investigation

text: of

text: traditional

text: front

text: are

text: described

text: by

text: the

text: hydrodynamic

text: laws

text: of

text: mass,

text: gaseous

text: combustion

text: detonations,

text: in

text: order

text: to

text: highlight

text: momentum

text: and

text: energy

text: conservation

text: expressed

text: in

text: terms

text: the

text: similarity

text: and

text: diﬀerence

text: between

text: the

text: combustion

text: and

text: of

text: magnetic

text: detonations.

text: By

text: deﬁnition,

text: detonation

text: is

text: a

text: fast

text: supersonic

text: combustion

text: regime,

text: for

text: which

text: preheating

text: of

text: the

text: cold

text: fuel

text: mixture

text: happens

text: due

text: to

text: the

text: leading

text: shock,

text: ρu

text: const

text: ρ

text: D,

text: see

text: Fig.

text: In

text: turn,

text: shock

text: propagation

text: without

text: decay

text: in

text: combustion

text: detonation

text: is

text: supported

text: by

text: energy

text: release

text: in

text: chemical

text: reactions

text: in

text: the

text: active

text: reaction

text: zone.

text: Typically

text: for

text: combustion,

text: the

text: activation

text: energy

text: of

text: the

text: reactions

text: is

text: quite

text: high,

text: so

text: that

text: reactions

text: develop

text: relatively

text: slow

text: just

text: P

text: const

text: P

text: ρ

text: after

text: the

text: leading

text: shock

text: in

text: the

text: so-called

text: induction

text: zone

text: (still,

text: much

text: faster

text: than

text: in

text: the

text: fuel

text: mixture).

text: It

text: requires

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: P/P 0 3 (cid:18) (cid:19) (cid:18) (cid:19) 45 u2 D2 40 s ρu h + = const = ρ D h + , (3) Shock 2 3 0 0 2 35 30 Detonation where h is the gas enthalpy 25 20 d γP 15 h = − + aQ, (4) (γ 1)ρ 10 5 γ is the adiabatic exponent, Q is the chemical energy O stored per unit mass, a is fraction of the unburned fuel 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 mixture, which changes from a = a = 1 in the fresh and 0 s shocked gas to a = 0 in the detonation products. The V/V d 0 hydrodynamic equations Eqs. (1)-(3) are complemented by the equation of state, which is taken for gaseous det- FIG. 3: The pressure-volume diagram for shock (blue curve) onations in the form of the ideal gas law and detonation (red curve) in a gas mixture. The scaled en- − ergy release is (γ 1)Q/γP V = 9, with the initial volume per 0 0 · ρ unit mass V0 = 6.33 m3/kg and initial pressure P0 = 1.33 104 P = RT, (5) Pa. The green straight line shows transition from the shock m point s to the combustion products d in the CJ detonation. where R is the ideal gas constant and m is the molar mass of the gas (for simplicity we assume here that the from the front at local sound speed.21 The CJ detonation molar mass does not change in the combustion process). propagates with the smallest speed possible for detona- Traditionally, the theory of shock≡waves and detonations tions for a ﬁxed energy release, and indicated by the green employs volume per unit mass V 1/ρ for analyzing the tangent line in Fig. 3. Thus, on the pressure-volume di- process.21 agram, the CJ detonation structure corresponds ﬁrst to Modiﬁcations of the gas parameters in a leading shock the jump from the point O (the initial state) to the point are described by Eqs. (1)-(3) with zero energy release s at the shock, and then the reaction development behind in the reaction within the discontinuous shock, as = 1. the shock along the green straight line from the point s Then Eqs. (1)-(3) may be reduced to the so-called Hugo- to the point d (detonation products). It may be shown niot equation, which speciﬁes all possible ﬁnite states of that the CJ detonation speed is determined by the energy the shocked gas Ps, Vs for a ﬁxed initial state P0, V0; the release in the combustion process.21 result is demonstrated by the blue solid curve in Fig. 3 The internal structure of the reaction zone in the det- for the initial state corresponding to the ideal gas at ini- onation front (transition from the point s to the point d tial volume·per unit mass V0 = 6.33 m3/kg and pressure in Fig. 3) is speciﬁed by reaction kinetics of a particular P0 = 1.33 104 Pa. For any particular ﬁnal state s at fuel mixture. Here we illustrate the detonation struc- the shock, equations (1), (2) relate the slope of the green ture for the simpliﬁed model of acetylene-air combustion straight line (sO) in Fig. 3 to the front propagation speed described by a single one-step Arrhenius reaction22 D as21 − ∂a − − ρ2D2 = Ps − P0 . (6) ∂t = Kρa exp( Ea/T), (7) 0 V V 0 s where K = 109 m3/kg s is a pre-exponential factor, In order to obtain the ﬁnal state of the detonation E /T = 29.3 is the scaled activation energy. Then the a 0 products we solve Eqs. (1)-(3) for the case of complete stationary detonation structure is determined by Eq. (7) burning ad = 0; the result is plotted by t−he solid red curve rewritten in the reference frame of the front as in Fig. 3 for the scaled energy release (γ 1)Q/(γP V ) = 0 0 9 corresponding to the acetylene-air combustion.22 The ∂a − red curve (for detonation) corresponds to higher pressure u = Kρa exp( E /T). (8) for the same volume than the blue curve (for a shock) due ∂z a to the energy release in a detonation front. Still, unlike a shock, detonation speed is not a free parameter, but By solving Eq. (8) together with Eqs. (1)-(3) we ﬁnd it is determined by particular boundary conditions. For the detonation front structure as shown in Fig. 4. For example, the case of a freely propagating detonation cor- obtaining the plots we have used the scaled energy re- − responds to the so-called Chapman-Jouguet (CJ) detona- lease (γ 1)Q/(γP V ) = 9, with the initial gas pressure 0 0 tion regime, with the detonation products moving away P = 0.627 MPa and temperature T = 293 K. Then by 0 0 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P/P

label: paragraph

font: default

label: equation

text: s

text: ρu

text: h

text: const

text: ρ

text: D

text: h

text: Shock

text: Detonation

text: where

text: h

text: is

text: the

text: gas

text: enthalpy

text: d

text: γP

text: h

text: aQ,

text: (γ

text: γ

text: is

text: the

text: adiabatic

text: exponent,

text: Q

text: is

text: the

text: chemical

text: energy

text: O

text: stored

text: per

text: unit

text: mass,

text: a

text: is

text: fraction

text: of

text: the

text: unburned

text: fuel

text: mixture,

text: which

text: changes

text: from

text: a

text: a

text: in

text: the

text: fresh

text: and

text: s

text: shocked

text: gas

text: to

text: a

text: in

text: the

text: detonation

text: products.

text: The

text: V/V

text: d

text: hydrodynamic

text: equations

text: Eqs.

text: are

text: complemented

text: by

text: the

text: equation

text: of

text: state,

text: which

text: is

text: taken

text: for

text: gaseous

text: det-

text: FIG.

text: The

text: pressure-volume

text: diagram

text: for

text: shock

text: (blue

text: curve)

text: onations

text: in

text: the

text: form

text: of

text: the

text: ideal

text: gas

text: law

text: and

text: detonation

text: (red

text: curve)

text: in

text: a

text: gas

text: mixture.

text: The

text: scaled

text: en-

text: ergy

text: release

text: is

text: (γ

text: V

text: with

text: the

text: initial

text: volume

text: per

text: ρ

text: unit

text: mass

text: and

text: initial

text: pressure

text: P

text: RT,

text: Pa.

text: The

text: green

text: straight

text: line

text: shows

text: transition

text: from

text: the

text: shock

text: m

text: point

text: s

text: to

text: the

text: combustion

text: products

text: d

text: in

text: the

text: CJ

text: detonation.

text: where

text: R

text: is

text: the

text: ideal

text: gas

text: constant

text: and

text: m

text: is

text: the

text: molar

text: mass

text: of

text: the

text: gas

text: (for

text: simplicity

text: we

text: assume

text: here

text: that

text: the

text: from

text: the

text: front

text: at

text: local

text: sound

text: The

text: CJ

text: detonation

text: molar

text: mass

text: does

text: not

text: change

text: in

text: the

text: combustion

text: process).

text: propagates

text: with

text: the

text: smallest

text: speed

text: possible

text: for

text: detona-

text: Traditionally,

text: the

text: theory

text: of

text: shock≡waves

text: and

text: detonations

text: tions

text: for

text: a

text: ﬁxed

text: energy

text: release,

text: and

text: indicated

text: by

text: the

text: green

text: employs

text: volume

text: per

text: unit

text: mass

text: V

text: for

text: analyzing

text: the

text: tangent

text: line

text: in

text: Fig.

text: Thus,

text: on

text: the

text: pressure-volume

text: di-

text: agram,

text: the

text: CJ

text: detonation

text: structure

text: corresponds

text: ﬁrst

text: to

text: Modiﬁcations

text: of

text: the

text: gas

text: parameters

text: in

text: a

text: leading

text: shock

text: the

text: jump

text: from

text: the

text: point

text: O

text: (the

text: initial

text: state)

text: to

text: the

text: point

text: are

text: described

text: by

text: Eqs.

text: with

text: zero

text: energy

text: release

text: s

text: at

text: the

text: shock,

text: and

text: then

text: the

text: reaction

text: development

text: behind

text: in

text: the

text: reaction

text: within

text: the

text: discontinuous

text: shock,

text: as

text: the

text: shock

text: along

text: the

text: green

text: straight

text: line

text: from

text: the

text: point

text: s

text: Then

text: Eqs.

text: may

text: be

text: reduced

text: to

text: the

text: so-called

text: Hugo-

text: to

text: the

text: point

text: d

text: (detonation

text: products).

text: It

text: may

text: be

text: shown

text: niot

text: equation,

text: which

text: speciﬁes

text: all

text: possible

text: ﬁnite

text: states

text: of

text: that

text: the

text: CJ

text: detonation

text: speed

text: is

text: determined

text: by

text: the

text: energy

text: the

text: shocked

text: gas

text: Ps,

text: Vs

text: for

text: a

text: ﬁxed

text: initial

text: state

text: the

text: release

text: in

text: the

text: combustion

text: result

text: is

text: demonstrated

text: by

text: the

text: blue

text: solid

text: curve

text: in

text: Fig.

text: The

text: internal

text: structure

text: of

text: the

text: reaction

text: zone

text: in

text: the

text: det-

text: for

text: the

text: initial

text: state

text: corresponding

text: to

text: the

text: ideal

text: gas

text: at

text: ini-

text: onation

text: front

text: (transition

text: from

text: the

text: point

text: s

text: to

text: the

text: point

text: d

text: tial

text: volume·per

text: unit

text: mass

text: and

text: pressure

text: in

text: Fig.

text: is

text: speciﬁed

text: by

text: reaction

text: kinetics

text: of

text: a

text: particular

text: Pa.

text: For

text: any

text: particular

text: ﬁnal

text: state

text: s

text: at

text: fuel

text: mixture.

text: Here

text: we

text: illustrate

text: the

text: detonation

text: struc-

text: the

text: shock,

text: equations

text: relate

text: the

text: slope

text: of

text: the

text: green

text: ture

text: for

text: the

text: simpliﬁed

text: model

text: of

text: acetylene-air

text: combustion

text: straight

text: line

text: (sO)

text: in

text: Fig.

text: to

text: the

text: front

text: propagation

text: speed

text: described

text: by

text: a

text: single

text: one-step

text: Arrhenius

text: D

text: ∂a

text: Ps

text: ∂t

text: Kρa

text: exp(

text: Ea/T),

text: V

text: V

text: s

text: where

text: K

text: s

text: is

text: a

text: pre-exponential

text: factor,

text: In

text: order

text: to

text: obtain

text: the

text: ﬁnal

text: state

text: of

text: the

text: detonation

text: E

text: /T

text: is

text: the

text: scaled

text: activation

text: energy.

text: Then

text: the

text: a

text: products

text: we

text: solve

text: Eqs.

text: for

text: the

text: case

text: of

text: complete

text: stationary

text: detonation

text: structure

text: is

text: determined

text: by

text: Eq.

text: burning

text: ad

text: the

text: result

text: is

text: plotted

text: by

text: t−he

text: solid

text: red

text: curve

text: rewritten

text: in

text: the

text: reference

text: frame

text: of

text: the

text: front

text: as

text: in

text: Fig.

text: for

text: the

text: scaled

text: energy

text: release

text: (γ

text: V

text: corresponding

text: to

text: the

text: acetylene-air

text: The

text: ∂a

text: red

text: curve

text: (for

text: detonation)

text: corresponds

text: to

text: higher

text: pressure

text: u

text: Kρa

text: exp(

text: E

text: /T).

text: for

text: the

text: same

text: volume

text: than

text: the

text: blue

text: curve

text: (for

text: a

text: shock)

text: due

text: ∂z

text: a

text: to

text: the

text: energy

text: release

text: in

text: a

text: detonation

text: front.

text: Still,

text: unlike

text: a

text: shock,

text: detonation

text: speed

text: is

text: not

text: a

text: free

text: parameter,

text: but

text: By

text: solving

text: Eq.

text: together

text: with

text: Eqs.

text: we

text: ﬁnd

text: it

text: is

text: determined

text: by

text: particular

text: boundary

text: conditions.

text: For

text: the

text: detonation

text: front

text: structure

text: as

text: shown

text: in

text: Fig.

text: For

text: example,

text: the

text: case

text: of

text: a

text: freely

text: propagating

text: detonation

text: cor-

text: obtaining

text: the

text: plots

text: we

text: have

text: used

text: the

text: scaled

text: energy

text: re-

text: responds

text: to

text: the

text: so-called

text: Chapman-Jouguet

text: (CJ)

text: detona-

text: lease

text: (γ

text: V

text: with

text: the

text: initial

text: gas

text: pressure

text: tion

text: regime,

text: with

text: the

text: detonation

text: products

text: moving

text: away

text: P

text: MPa

text: and

text: temperature

text: T

text: K.

text: Then

text: by

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 B. The analytical theory for the key magnetic Expanding Eq. (20) in powers of δ and taking a = 0 detonation parameters we obtain pressu(cid:20)re in the(cid:18)detona(cid:19)t(cid:21)ion products as As we can see in Fig. 7, the limit of weak compression − (cid:28) P (cid:20) (cid:18) Γ (cid:19) (cid:21) (r 1) 1 holds with a very good accuracy for magnetic d =ΓQ + c2 + Γ + 1 δ ρ 0 2 d detonations, and hence allows developing the analytical 0 theory of the process. The system Eqs. (9)-(11) may be 1 Γ2 Γ − + Q + 1 + c2(n 1) δ2. (22) reduced to a single equation, which is equivalent to the 2 2 2 0 d Hugoniot relation21 with thermal conduction taken into account The scaled density deviation δ corresponding to the d CJ point of the detonation products on Fig. 7 is unknown − − − 1 r 1 P + P − κ dT so far and we have to ﬁnd it. At the CJ point the tangent ε ε 0 + Q(a 1) = . (18) 0 2 r ρ ρ D dz line touches the detonation curve, which implies 0 0 By combining the equations of state for the crystals P =P , (23) t d of nanomagnets, Eqs. (15), (16), we get rid of the ther- mal terms and obtain the expression for thermodynamic energy as function of P and r only ∂P ∂P 1 P (cid:20) (cid:18) (cid:19) (cid:21) t = d . (24) ε = ∂r ∂r rΓ ρ 0 c2 − 1 − 1 − n − (cid:28) + 0 (rn 1) − − (r 1) . For small compression case δ 1 we rewrite Eq. (17) nr n 1 Γ n 1 as (19) − P P − Then substituting Eq. (21) into the Hugonoit relation t 0 = D2δ(1 δ), (25) ρ Eq. (18) we obtain pressure as 0 Substituting Eqs.(22) and (25) into Eq.(23), (24), we P (cid:18) 2Γ { (cid:19) − = − − rQ(1 a) ﬁnd (cid:20) (cid:18) (cid:19)(cid:21) ρ 2 Γ(1 r) 0 − r(cid:20) 1 κ dT(cid:18) (cid:19)(cid:21)(cid:27) + Γ + r ε + r − (cid:20) (cid:18) Γ(cid:19) (cid:21) 2 0 ρ D dz D2δ (1 δ ) =ΓQ + c2 + ΓQ + 1 δ 0 d d 0 2 d c2 n − − − 1 − 1 + 0 − (r 1) (rn 1) − . 1 Γ2 Γ − n n 1 n 1 Γ + Q + 1 + c2(n 1) δ2, 2 2 2 0 d (20) (26) (cid:18) (cid:19) In a similar way, by substituting Eqs. (15), (16) into Eq. (18) we ﬁnd temperature − (cid:20) (cid:18)Γ (cid:19) (cid:21) D2(1 2δ ) =c2 + ΓQ + 1 2(α +(cid:20) 1)MΘα { (cid:18)− (cid:19)(cid:21) d 0 2 Tα+1 = − −D Q(1 a) RA [2 Γ(1 r)] Γ2 Γ − − + Q + 1 + c2(n 1) δ . (27) (cid:18)c2 n −(cid:19) − 1 (cid:27)− r 1 2 2 0 d + 0 − (r 1) − nr n 1 n 1 2 − r 1 κ dT Then, eliminating D∝2 from Eqs. (26), (27), and keep- + 2r Γ + 1 ε0 + ρ D dz . (21) ing terms as small as δd2 we obtain 0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) − Now we take into account that density variations in ΓQ (1 2Γδ ) = d the magnetic detonation are small by introducing a small − − (cid:28) Γ Γ 1 vinailtuiaelδth=ermr od1yn=amρi/cρe0nerg1y sin1c.e Wε e/Qca∼n 1a0ls−o6.neFgilresctt, +δd2 ΓQ 1 + 2 1 + 4 + 2c20(n + 1) 0 → ∞ (28) we analyze the detonation products, z , for which → the transformation process is completed, dT/dz 0. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: B.

label: section

text: The

text: analytical

text: theory

text: for

text: the

text: key

text: magnetic

text: Expanding

text: Eq.

text: in

text: powers

text: of

text: δ

text: and

text: taking

text: a

text: detonation

text: parameters

text: we

text: obtain

text: in

text: products

text: as

text: As

text: we

text: can

text: see

text: in

text: Fig.

text: the

text: limit

text: of

text: weak

text: compression

text: P

text: Γ

text: (r

text: holds

text: with

text: a

text: very

text: good

text: accuracy

text: for

text: magnetic

text: d

text: =ΓQ

text: Γ

text: δ

text: ρ

text: d

text: detonations,

text: and

text: hence

text: allows

text: developing

text: the

text: analytical

text: theory

text: of

text: the

text: process.

text: The

text: system

text: Eqs.

text: may

text: be

text: Γ2

text: Γ

text: Q

text: reduced

text: to

text: a

text: single

text: equation,

text: which

text: is

text: equivalent

text: to

text: the

text: d

text: Hugoniot

text: with

text: thermal

text: conduction

text: taken

text: into

text: account

text: The

text: scaled

text: density

text: deviation

text: δ

text: corresponding

text: to

text: the

text: d

text: CJ

text: point

text: of

text: the

text: detonation

text: products

text: on

text: Fig.

text: is

text: unknown

text: r

text: P

text: P

text: κ

text: dT

text: so

text: far

text: and

text: we

text: have

text: to

text: ﬁnd

text: it.

text: At

text: the

text: CJ

text: point

text: the

text: tangent

text: ε

text: ε

text: Q(a

text: r

text: ρ

text: ρ

text: D

text: dz

text: line

text: touches

text: the

text: detonation

text: curve,

text: which

text: implies

text: By

text: combining

text: the

text: equations

text: of

text: state

text: for

text: the

text: crystals

text: P

text: =P

text: t

text: d

text: of

text: nanomagnets,

text: Eqs.

text: we

text: get

text: rid

text: of

text: the

text: ther-

text: mal

text: terms

text: and

text: obtain

text: the

text: expression

text: for

text: thermodynamic

text: energy

text: as

text: function

text: of

text: P

text: and

text: r

text: only

text: ∂P

text: ∂P

text: P

text: t

text: d

text: ε

text: ∂r

text: ∂r

text: rΓ

text: ρ

text: n

text: (rn

text: (r

text: For

text: small

text: compression

text: case

text: δ

text: we

text: rewrite

text: Eq.

text: nr

text: n

text: Γ

text: n

text: as

text: P

text: P

text: Then

text: substituting

text: Eq.

text: into

text: the

text: Hugonoit

text: relation

text: t

text: δ),

text: ρ

text: Eq.

text: we

text: obtain

text: pressure

text: as

text: Substituting

text: and

text: into

text: we

text: P

text: 2Γ

text: a)

text: ﬁnd

text: ρ

text: r)

text: κ

text: Γ

text: r

text: ε

text: r

text: ρ

text: D

text: dz

text: δ

text: =ΓQ

text: ΓQ

text: δ

text: d

text: d

text: d

text: n

text: (r

text: (rn

text: Γ2

text: Γ

text: n

text: n

text: n

text: Γ

text: Q

text: d

text: In

text: a

text: similar

text: way,

text: by

text: substituting

text: Eqs.

text: into

text: Eq.

text: we

text: ﬁnd

text: temperature

text: 2δ

text: ΓQ

text: d

text: −D

text: a)

text: RA

text: r)]

text: Γ2

text: Γ

text: Q

text: δ

text: n

text: r

text: d

text: (r

text: nr

text: n

text: n

text: r

text: κ

text: dT

text: Then,

text: eliminating

text: from

text: Eqs.

text: and

text: keep-

text: Γ

text: ε0

text: ρ

text: D

text: dz

text: ing

text: terms

text: as

text: small

text: as

text: we

text: obtain

text: Now

text: we

text: take

text: into

text: account

text: that

text: density

text: variations

text: in

text: ΓQ

text: 2Γδ

text: d

text: the

text: magnetic

text: detonation

text: are

text: small

text: by

text: introducing

text: a

text: small

text: Γ

text: Γ

text: vinailtuiaelδth=ermr

text: Wε

text: e/Qca∼n

text: ΓQ

text: we

text: analyze

text: the

text: detonation

text: products,

text: z

text: for

text: which

text: the

text: transformation

text: process

text: is

text: completed,

text: dT/dz

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: )aP( P-P t a , T/T , P/P )aPk( P d s 10 1000 1.2 800 a T/T 600 1 d 400 0.8 (cid:536)'=0.005 CJ point 200 Initial point 0 0.6 P/P Shock s -200 Spin flipping (cid:536)'=0.01 0.4 -400 -600 0.2 -800 (cid:536)'=0.0025 -1000 0 0.98 0.985 0.99 0.995 1 0 10 20 30 40 v/v z/L 0 0 FIG. 11: The pressure-volume diagram for diﬀerent values FIG. 12: Proﬁles of scaled pressure, temperature and fraction (cid:48) of the scaled viscosity η = 0.0025; 0.005; 0.01 in the external of nanomagnets in the metastable state for magnetic detona- (cid:48) magnetic ﬁeld B = 4T. tion for scaled viscosity η = 0.005 in the external magnetic z ﬁeld B = 4T. z strongly a discontinuous shock wave; this pressure in- 100 crease is followed by pressure relaxation to a smaller (cid:536)'=0.005 (cid:536)'=0.0025 value in the process of spin-ﬂipping with Zeeman energy 80 release. As we take larger values of volume viscosity, (cid:48) η = 0.005; 0.01, deviation of the pressure-volume plots Shock 60 from the discontinuous shock model becomes much more Spin flipping pronounced, Fig. 11, and we ﬁnd pressure and density (cid:536)'=0.01 in the shock wave decreasing as compared to the discon- 40 tinuous case, Fig. 13. In fact, the very deﬁnition of a shock as a part of magnetic detonation front becomes 20 ambiguous when volume viscosity is taken into account. To avoid the ambiguity, we notice that within the model 0 of a discontinuous shock, pressure maximum is attained -20 -10 0 10 20 30 40 at the shock front, and then pressure goes down as the z/L 0 spin-ﬂipping starts. In the same way it seems natural treating the point of maximal pressure in Figs. 12, 13 as FIG. 13: Pressure proﬁles in magnetic detonation for diﬀer- the back-side of the shock region. In Fig. 13 we placed (cid:48) ent values of the scaled viscosity η = 0.0025; 0.005; 0.01 in all three plots by choosing z = 0 as the position of the the external magnetic ﬁeld B = 4T. Pressure maxima are z pressure maximum and hence the back side of the shocks. placed at the position z = 0, which may be deﬁned as the Then the regions corresponding to z < 0 belong to the back side of the shock region in the detonation. shocks smoothed by viscosity, while the domain of z > 0 may be treated roughly as the regions of spin-ﬂipping ∼ with Zeeman energy release. Of course, such sep- front does not change much, being about 10L for all − 0 aration is rather qualitative than quantitative, because three cases, i.e. about 1 2 mm in dimensional units. in presence of considerable volume viscosity spin-ﬂipping We point out that the experimentally employed sam- starts already in the shock, see Fig. 12 corresponding to ple sizes for the crystals of nanomagnets are also about 2 (cid:48) η = 0.005, B = 4T. In Fig. 12 the fraction of nanomag- mm, e.g. see Refs. [1,7]. As a result, it is rather diﬃcult z ≈ nets in the metastable state is about a 0.6 at the point to observe steady well-developed magnetic detonation in of pressure maximum. Figure 14 compares temperature the commonly experimental conditions. Instead, we sug- (cid:48) proﬁles for the scaled viscosity η = 0.0025; 0.005; 0.01 gest that most of the experimental points reported in Ref. with the respective pressure maxima still placed at z = 0. [1] for ultra-fast magnetic avalanches correspond to mag- As we can see, larger volume viscosity increases strongly netic detonation in the process of development, which (cid:48) temperature at the shock; in the case of η = 0.01 tem- is also indicated by the average avalanche speed in the perature at z = 0 is only 5% smaller than the ﬁnal det- samples noticeably below the sound speed. Another im- ∼ onation temperature, i.e. it is about 0.95T . Tem- portant feature of the experimental observations of Ref. d perature increase reduces strongly the size of the ”pure” [1] is that the ultra-fast avalanches were obtained for the spin-ﬂipping region, still, as we can see from Fig. 14, it magnetic ﬁeld close to the quantum resonance values of is accompanied by strong increase of the width of the the nanomagnets. Quantum resonances lead to strong shock wave, so that total width of magnetic detonation decrease of the factor τ in the kinetic equation of spin- ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: )aP(

text: P-P

text: t

text: a

text: T/T

text: P/P

text: )aPk(

text: P

text: d

text: s

text: a

text: T/T

text: d

text: CJ

text: point

text: Initial

text: point

text: P/P

text: Shock

text: s

text: Spin

text: flipping

text: v/v

text: z/L

text: FIG.

text: The

text: pressure-volume

text: diagram

text: for

text: diﬀerent

text: values

text: FIG.

text: Proﬁles

text: of

text: scaled

text: pressure,

text: temperature

text: and

text: fraction

text: of

text: the

text: scaled

text: viscosity

text: η

text: in

text: the

text: external

text: of

text: nanomagnets

text: in

text: the

text: metastable

text: state

text: for

text: magnetic

text: detona-

text: magnetic

text: ﬁeld

text: B

text: tion

text: for

text: scaled

text: viscosity

text: η

text: in

text: the

text: external

text: magnetic

text: z

text: ﬁeld

text: B

text: z

text: strongly

text: a

text: discontinuous

text: shock

text: wave;

text: this

text: pressure

text: in-

text: crease

text: is

text: followed

text: by

text: pressure

text: relaxation

text: to

text: a

text: smaller

text: value

text: in

text: the

text: process

text: of

text: spin-ﬂipping

text: with

text: Zeeman

text: energy

text: release.

text: As

text: we

text: take

text: larger

text: values

text: of

text: volume

text: viscosity,

text: η

text: deviation

text: of

text: the

text: pressure-volume

text: plots

text: Shock

text: from

text: the

text: discontinuous

text: shock

text: model

text: becomes

text: much

text: more

text: Spin

text: flipping

text: pronounced,

text: Fig.

text: and

text: we

text: ﬁnd

text: pressure

text: and

text: density

text: in

text: the

text: shock

text: wave

text: decreasing

text: as

text: compared

text: to

text: the

text: discon-

text: tinuous

text: case,

text: Fig.

text: In

text: fact,

text: the

text: very

text: deﬁnition

text: of

text: a

text: shock

text: as

text: a

text: part

text: of

text: magnetic

text: detonation

text: front

text: becomes

text: ambiguous

text: when

text: volume

text: viscosity

text: is

text: taken

text: into

text: account.

text: To

text: avoid

text: the

text: ambiguity,

text: we

text: notice

text: that

text: within

text: the

text: model

text: of

text: a

text: discontinuous

text: shock,

text: pressure

text: maximum

text: is

text: attained

text: at

text: the

text: shock

text: front,

text: and

text: then

text: pressure

text: goes

text: down

text: as

text: the

text: z/L

text: spin-ﬂipping

text: starts.

text: In

text: the

text: same

text: way

text: it

text: seems

text: natural

text: treating

text: the

text: point

text: of

text: maximal

text: pressure

text: in

text: Figs.

text: as

text: FIG.

text: Pressure

text: proﬁles

text: in

text: magnetic

text: detonation

text: for

text: diﬀer-

text: the

text: back-side

text: of

text: the

text: shock

text: region.

text: In

text: Fig.

text: we

text: placed

text: ent

text: values

text: of

text: the

text: scaled

text: viscosity

text: η

text: in

text: all

text: three

text: plots

text: by

text: choosing

text: z

text: as

text: the

text: position

text: of

text: the

text: the

text: external

text: magnetic

text: ﬁeld

text: B

text: Pressure

text: maxima

text: are

text: z

text: pressure

text: maximum

text: and

text: hence

text: the

text: back

text: side

text: of

text: the

text: shocks.

text: placed

text: at

text: the

text: position

text: z

text: which

text: may

text: be

text: deﬁned

text: as

text: the

text: Then

text: the

text: regions

text: corresponding

text: to

text: z

text: belong

text: to

text: the

text: back

text: side

text: of

text: the

text: shock

text: region

text: in

text: the

text: detonation.

text: shocks

text: smoothed

text: by

text: viscosity,

text: while

text: the

text: domain

text: of

text: z

text: may

text: be

text: treated

text: roughly

text: as

text: the

text: regions

text: of

text: spin-ﬂipping

text: with

text: Zeeman

text: energy

text: release.

text: Of

text: course,

text: such

text: sep-

text: front

text: does

text: not

text: change

text: much,

text: being

text: about

text: for

text: all

text: aration

text: is

text: rather

text: qualitative

text: than

text: quantitative,

text: because

text: three

text: cases,

text: i.e.

text: about

text: mm

text: in

text: dimensional

text: units.

text: in

text: presence

text: of

text: considerable

text: volume

text: viscosity

text: spin-ﬂipping

text: We

text: point

text: out

text: that

text: the

text: experimentally

text: employed

text: sam-

text: starts

text: already

text: in

text: the

text: shock,

text: see

text: Fig.

text: corresponding

text: to

text: ple

text: sizes

text: for

text: the

text: crystals

text: of

text: nanomagnets

text: are

text: also

text: about

text: η

text: B

text: In

text: Fig.

text: the

text: fraction

text: of

text: nanomag-

text: mm,

text: e.g.

text: see

text: Refs.

text: As

text: a

text: result,

text: it

text: is

text: rather

text: diﬃcult

text: z

text: nets

text: in

text: the

text: metastable

text: state

text: is

text: about

text: a

text: at

text: the

text: point

text: to

text: observe

text: steady

text: well-developed

text: magnetic

text: detonation

text: in

text: of

text: pressure

text: maximum.

text: Figure

text: compares

text: temperature

text: the

text: commonly

text: experimental

text: conditions.

text: Instead,

text: we

text: sug-

text: proﬁles

text: for

text: the

text: scaled

text: viscosity

text: η

text: gest

text: that

text: most

text: of

text: the

text: experimental

text: points

text: reported

text: in

text: Ref.

text: with

text: the

text: respective

text: pressure

text: maxima

text: still

text: placed

text: at

text: z

text: for

text: ultra-fast

text: magnetic

text: avalanches

text: correspond

text: to

text: mag-

text: As

text: we

text: can

text: see,

text: larger

text: volume

text: viscosity

text: increases

text: strongly

text: netic

text: detonation

text: in

text: the

text: process

text: of

text: development,

text: which

text: temperature

text: at

text: the

text: shock;

text: in

text: the

text: case

text: of

text: η

text: tem-

text: is

text: also

text: indicated

text: by

text: the

text: average

text: avalanche

text: speed

text: in

text: the

text: perature

text: at

text: z

text: is

text: only

text: smaller

text: than

text: the

text: ﬁnal

text: det-

text: samples

text: noticeably

text: below

text: the

text: sound

text: speed.

text: Another

text: im-

text: onation

text: temperature,

text: i.e.

text: it

text: is

text: about

text: Tem-

text: portant

text: feature

text: of

text: the

text: experimental

text: observations

text: of

text: Ref.

text: d

text: perature

text: increase

text: reduces

text: strongly

text: the

text: size

text: of

text: the

text: ”pure”

text: is

text: that

text: the

text: ultra-fast

text: avalanches

text: were

text: obtained

text: for

text: the

text: spin-ﬂipping

text: region,

text: still,

text: as

text: we

text: can

text: see

text: from

text: Fig.

text: it

text: magnetic

text: ﬁeld

text: close

text: to

text: the

text: quantum

text: resonance

text: values

text: of

text: is

text: accompanied

text: by

text: strong

text: increase

text: of

text: the

text: width

text: of

text: the

text: the

text: nanomagnets.

text: Quantum

text: resonances

text: lead

text: to

text: strong

text: shock

text: wave,

text: so

text: that

text: total

text: width

text: of

text: magnetic

text: detonation

text: decrease

text: of

text: the

text: factor

text: τ

text: in

text: the

text: kinetic

text: equation

text: of

text: spin-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: A PERSISTENCE LANDSCAPES TOOLBOX FOR TOPOLOGICAL STATISTICS PETER BUBENIK AND PAWEL(cid:32) DL(cid:32) OTKO Abstract. Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning. We give eﬃcient algorithms for calculating persistence landscapes, their averages, and distances between such averages. We discuss an implementation of these algorithms and some related procedures. These are intended to facilitate the combination of statistics and machine learning with topological data analysis. We present an experiment showing that the low-dimensional persistence landscapes of points sampled from spheres (and boxes) of varying dimensions diﬀer. 1. Introduction We provide some algorithms and computational tools for statistical topological data analysis. In particular, we give algorithms for calculating the persistence landscape, a functional summary of persistence modules. We also give algorithms for calculating the averages of such summaries, and for calculating distances between such averages. These tools also provide an alternative com- putational approach for calculating distances between topological summaries that may be useful when other methods are computational prohibitive. In addition, we specify an implementation of these algorithms and some related tools that we have made publicly available. We are motivated by topological data analysis [30, 8]. Its main tool, persistent homology pro- vides a multiscale description of the topology of the data of interest, called either a barcode or a persistence diagram. Unfortunately this summary is diﬃcult to work with from the point of view of statistics and machine learning. For example, it is not feasible to calculate averages. For these purposes, it is convenient to replace these summaries with a linear summary, that is, a ﬁnite- or inﬁnite-dimensional vector. In a linear space it is easy to calculate averages. One such vector which does not lose any information is the functional summary called the persistence landscape [6]. Since this summary may be thought of as lying in a Hilbert space, in the language of machine learning, it is a feature map. There is an associated kernel [42] to which standard machine learning tools may be applied. In the simplest computational setting for topological data analysis, the data of interest is encoded in a ﬁnite ﬁltered complex, K ⊂ K ⊂ ⊂ K (1) . . . . 0 1 n ∅ This is a ﬁltration of the complex K = K and it is sometime convenient to add K− = . n 1 Persistent homology [24, 49] gives a multiscale representation of the topology of this complex. To Key words and phrases. topological data analysis, persistent homology, statistical topology, topological machine learning, intrinsic dimension. P.B is supported by AFOSR grant FA9550-13-1-0115. P.D is supported by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions), DARPA grant FA9550-12-1-0416 and AFOSR grant FA9550-14-1-0012. 1 ##LTLine##

text: A

text: PERSISTENCE

text: LANDSCAPES

text: TOOLBOX

text: FOR

text: TOPOLOGICAL

text: STATISTICS

text: PETER

text: BUBENIK

text: AND

text: OTKO

text: Abstract.

text: Topological

label: abstract

text: data

text: analysis

text: provides

text: a

text: multiscale

text: description

text: of

text: the

text: geometry

text: and

text: topology

text: of

text: quantitative

text: data.

text: The

text: persistence

text: landscape

text: is

text: a

text: topological

text: summary

text: that

text: can

text: be

text: easily

text: combined

text: with

text: tools

text: from

text: statistics

text: and

text: machine

text: learning.

text: We

text: give

text: eﬃcient

text: algorithms

text: for

text: calculating

text: persistence

text: landscapes,

text: their

text: averages,

text: and

text: distances

text: between

text: such

text: averages.

text: We

text: discuss

text: an

text: implementation

text: of

text: these

text: algorithms

text: and

text: some

text: related

text: procedures.

text: These

text: are

text: intended

text: to

text: facilitate

text: the

text: combination

text: of

text: statistics

text: and

text: machine

text: learning

text: with

text: topological

text: data

text: analysis.

text: We

text: present

text: an

text: experiment

text: showing

text: that

text: the

text: low-dimensional

text: persistence

text: landscapes

text: of

text: points

text: sampled

text: from

text: spheres

text: (and

text: boxes)

text: of

text: varying

text: dimensions

text: diﬀer.

text: Introduction

text: We

text: provide

text: some

text: algorithms

text: and

text: computational

text: tools

text: for

text: statistical

text: topological

text: data

text: analysis.

text: In

text: particular,

text: we

text: give

text: algorithms

text: for

text: calculating

text: the

text: persistence

text: landscape,

text: a

text: functional

text: summary

text: of

text: persistence

text: modules.

text: We

text: also

text: give

text: algorithms

text: for

text: calculating

text: the

text: averages

text: of

text: such

text: summaries,

text: and

text: for

text: calculating

text: distances

text: between

text: such

text: averages.

text: These

text: tools

text: also

text: provide

text: an

text: alternative

text: com-

text: putational

text: approach

text: for

text: calculating

text: distances

text: between

text: topological

text: summaries

text: that

text: may

text: be

text: useful

text: when

text: other

text: methods

text: are

text: computational

text: prohibitive.

text: In

text: addition,

text: we

text: specify

text: an

text: implementation

text: of

text: these

text: algorithms

text: and

text: some

text: related

text: tools

text: that

text: we

text: have

text: made

text: publicly

text: available.

text: We

text: are

text: motivated

text: by

text: topological

text: data

text: analysis

text: Its

text: main

text: tool,

text: persistent

text: homology

text: pro-

text: vides

text: a

text: multiscale

text: description

text: of

text: the

text: topology

text: of

text: the

text: data

text: of

text: interest,

text: called

text: either

text: a

text: barcode

text: or

text: a

text: persistence

text: diagram.

text: Unfortunately

text: this

text: summary

text: is

text: diﬃcult

text: to

text: work

text: with

text: from

text: the

text: point

text: of

text: view

text: of

text: statistics

text: and

text: machine

text: learning.

text: For

text: example,

text: it

text: is

text: not

text: feasible

text: to

text: calculate

text: averages.

text: For

text: these

text: purposes,

text: it

text: is

text: convenient

text: to

text: replace

text: these

text: summaries

text: with

text: a

text: linear

text: summary,

text: that

text: is,

text: a

text: ﬁnite-

text: or

text: inﬁnite-dimensional

text: vector.

text: In

text: a

text: linear

text: space

text: it

text: is

text: easy

text: to

text: calculate

text: averages.

text: One

text: such

text: vector

text: which

text: does

text: not

text: lose

text: any

text: information

text: is

text: the

text: functional

text: summary

text: called

text: the

text: persistence

text: landscape

text: Since

text: this

text: summary

text: may

text: be

text: thought

text: of

text: as

text: lying

text: in

text: a

text: Hilbert

text: space,

text: in

text: the

text: language

text: of

text: machine

text: learning,

text: it

text: is

text: a

text: feature

text: map.

text: There

text: is

text: an

text: associated

text: kernel

text: to

text: which

text: standard

text: machine

text: learning

text: tools

text: may

text: be

text: applied.

text: In

text: the

text: simplest

text: computational

text: setting

text: for

text: topological

text: data

text: analysis,

text: the

text: data

text: of

text: interest

text: is

text: encoded

text: in

text: a

text: ﬁnite

text: ﬁltered

text: complex,

text: K

text: K

text: K

text: n

text: This

text: is

text: a

text: ﬁltration

text: of

text: the

text: complex

text: K

text: K

text: and

text: it

text: is

text: sometime

text: convenient

text: to

text: add

text: K−

text: n

text: Persistent

text: homology

text: gives

text: a

text: multiscale

text: representation

text: of

text: the

text: topology

text: of

text: this

text: complex.

text: To

text: Key

text: words

text: and

text: phrases.

text: topological

text: data

text: analysis,

text: persistent

text: homology,

text: statistical

text: topology,

text: topological

text: machine

text: learning,

text: intrinsic

text: dimension.

text: P.B

text: is

text: supported

text: by

text: AFOSR

text: grant

text: P.D

text: is

text: supported

text: by

text: the

text: Advanced

text: Grant

text: of

text: the

text: European

text: Research

text: Council

text: GUDHI

text: (Geometric

text: Understanding

text: in

text: Higher

text: Dimensions),

text: DARPA

text: grant

text: and

text: AFOSR

text: grant

text: ##LTLine##


text: A PERSISTENCE LANDSCAPES TOOLBOX FOR TOPOLOGICAL STATISTICS 11 ∈ { } For d 2, 3, . . . , 10 , we sampled 100 points from Sd using the uniform distribution. This − was done by sampling 100 points from the (d + 1) dimensional Gaussian distribution and then projecting those points to Sd. For an example of such normalization in topological data analysis, see [9]. This was repeated 1000 times. To refute the charge that our detected diﬀerences are only due to the increase of the average distance between points in higher dimensions, we rescale the sphere on which the points has been projected so that the average distance between points is one. For each point cloud generated as described above we computed the persistent homology of the corresponding Vietoris-Rips complex. The parameter values for the radius ranged from 0 to a radius for which all inessential 0, 1, and 2 dimensional cycles are killed, which was 0.7 for this range of dimensions. Remark 2. In this example and all subsequent examples, the ﬁltration values have been rescaled to the range from 0 to 100. For simplicity, we have chosen to leave all subsequent calculation in this new scale. If desired, it is easy to rescale the results back to the original scale. The resulting average persistence landscapes for degree zero, one and two are in Figure 3, Fig- ∞ ure 4, and Figure 5, respectively. The L1, L2 and L distances between these average landscapes given in Figure 6. To determine the signiﬁcance of these distances between average landscapes we performed a permutation test [47], which we now explain. First choose a signiﬁcance level α = 0.05. For ∈ { } (cid:54) every pair i, j 2, . . . , 10 with i = j, the two corresponding sets of 1000 persistence landscapes were combined in a set of cardinality 2000. Then this set was randomly split into two subsets A , A of cardinality 1000 each. Next the average landscapes λ and λ were computed based on 1 2 1 2 the landscapes in A and A , respectively. Let δ denote the distance between the original average 1 2 landscapes in dimensions i and j. The distance between λ and λ is compared to δ. This described 1 2 process is repeated 10, 000 times. The p value equals the proportion of cases in which the distance (cid:54) between λ and λ is greater than δ. For every pair i = j, it never happened that the distance 1 2 between λ and λ was greater than the corresponding δ. Therefore we can conclude that there is 1 2 very strong statistical diﬀerence between the persistence landscapes in various dimensions. We believe that the ability to easily perform such calculations will be useful in topological data analysis. Given the success of the experiment presented in the Section it is natural to ask analogous question for the case of points sampled from a d-dimensional box [0, 1]d. Theorems on the random Vietoris-Rips and Cˇech complexes can be found in [33]. The results presented there indicate that Betti numbers of random complexes “do not live together”; see Figure 1 in [33]. Here we focus not on the Betti numbers and their limit distribution, but on a task of dimension detection based on persistent homology. ∈ { } For d 2, 3, . . . , 10 , to get a single point cloud, we sampled 100 points from [0, 1]d by sampling each coordinate of each of the points independently from an interval [0, 1]. For each dimension ∈ { } d 2, 3, . . . , 10 , we sampled 1000 point clouds. As in the previous section, to refute the charge that our detected diﬀerences are only due to the increase of the average distance between points in higher dimensions, we rescaled the boxes so that the average distance between points in each box is 1. We computed persistent homology in dimension 0 and 1 of the resulting Rips complexes. We ∈ { } obtained 1000 persistence diagrams in dimension 0 and 1 for each dimension d 2, 3, . . . , 10 . The permutation test was run for 10, 000 permutations for the obtained persistence intervals separately in dimension 0 and in dimension 1. In all cases, it never happened that the distance between shuﬄed sets was greater than between the original ones. This experiment, together with the one presented in the Section , show that low dimensional persistent homology can be a good tool for estimating the intrinsic dimension of data sets. Knowing

text: A

text: PERSISTENCE

text: LANDSCAPES

text: TOOLBOX

text: FOR

text: TOPOLOGICAL

text: STATISTICS

text: For

text: d

text: we

text: sampled

text: points

text: from

text: Sd

text: using

text: the

text: uniform

text: distribution.

text: This

text: was

text: done

text: by

text: sampling

text: points

text: from

text: the

text: (d

text: dimensional

text: Gaussian

text: distribution

text: and

text: then

text: projecting

text: those

text: points

text: to

text: Sd.

text: For

text: an

text: example

text: of

text: such

text: normalization

text: in

text: topological

text: data

text: analysis,

text: see

text: This

text: was

text: repeated

text: times.

text: To

text: refute

text: the

text: charge

text: that

text: our

text: detected

text: diﬀerences

text: are

text: only

text: due

text: to

text: the

text: increase

text: of

text: the

text: average

text: distance

text: between

text: points

text: in

text: higher

text: dimensions,

text: we

text: rescale

text: the

text: sphere

text: on

text: which

text: the

text: points

text: has

text: been

text: projected

text: so

text: that

text: the

text: average

text: distance

text: between

text: points

text: is

text: one.

text: For

text: each

text: point

text: cloud

text: generated

text: as

text: described

text: above

text: we

text: computed

text: the

text: persistent

text: homology

text: of

text: the

text: corresponding

text: Vietoris-Rips

text: complex.

text: The

text: parameter

text: values

text: for

text: the

text: radius

text: ranged

text: from

text: to

text: a

text: radius

text: for

text: which

text: all

text: inessential

text: and

text: dimensional

text: cycles

text: are

text: killed,

text: which

text: was

text: for

text: this

text: range

text: of

text: dimensions.

text: Remark

text: In

text: this

text: example

text: and

text: all

text: subsequent

text: examples,

text: the

text: ﬁltration

text: values

text: have

text: been

text: rescaled

text: to

text: the

text: range

text: from

text: to

text: For

text: simplicity,

text: we

text: have

text: chosen

text: to

text: leave

text: all

text: subsequent

text: calculation

text: in

text: this

text: new

text: scale.

text: If

text: desired,

text: it

text: is

text: easy

text: to

text: rescale

text: the

text: results

text: back

text: to

text: the

text: original

text: scale.

text: The

text: resulting

text: average

text: persistence

text: landscapes

text: for

text: degree

text: zero,

text: one

text: and

text: two

text: are

text: in

text: Figure

text: Fig-

text: ure

text: and

text: Figure

text: respectively.

text: The

text: and

text: L

text: distances

text: between

text: these

text: average

text: landscapes

text: given

text: in

text: Figure

text: To

text: determine

text: the

text: signiﬁcance

text: of

text: these

text: distances

text: between

text: average

text: landscapes

text: we

text: performed

text: a

text: permutation

text: test

text: which

text: we

text: now

text: explain.

text: First

text: choose

text: a

text: signiﬁcance

text: level

text: α

text: For

text: every

text: pair

text: i,

text: j

text: with

text: i

text: j,

text: the

text: two

text: corresponding

text: sets

text: of

text: persistence

text: landscapes

text: were

text: combined

text: in

text: a

text: set

text: of

text: cardinality

text: Then

text: this

text: set

text: was

text: randomly

text: split

text: into

text: two

text: subsets

text: A

text: A

text: of

text: cardinality

text: each.

text: Next

text: the

text: average

text: landscapes

text: λ

text: and

text: λ

text: were

text: computed

text: based

text: on

text: the

text: landscapes

text: in

text: A

text: and

text: A

text: respectively.

text: Let

text: δ

text: denote

text: the

text: distance

text: between

text: the

text: original

text: average

text: landscapes

text: in

text: dimensions

text: i

text: and

text: j.

text: The

text: distance

text: between

text: λ

text: and

text: λ

text: is

text: compared

text: to

text: δ.

text: This

text: described

text: process

text: is

text: repeated

text: times.

text: The

text: p

text: value

text: equals

text: the

text: proportion

text: of

text: cases

text: in

text: which

text: the

text: distance

text: between

text: λ

text: and

text: λ

text: is

text: greater

text: than

text: δ.

text: For

text: every

text: pair

text: i

text: j,

text: it

text: never

text: happened

text: that

text: the

text: distance

text: between

text: λ

text: and

text: λ

text: was

text: greater

text: than

text: the

text: corresponding

text: δ.

text: Therefore

text: we

text: can

text: conclude

text: that

text: there

text: is

text: very

text: strong

text: statistical

text: diﬀerence

text: between

text: the

text: persistence

text: landscapes

text: in

text: various

text: dimensions.

text: We

text: believe

text: that

text: the

text: ability

text: to

text: easily

text: perform

text: such

text: calculations

text: will

text: be

text: useful

text: in

text: topological

text: data

text: analysis.

text: Given

text: the

text: success

text: of

text: the

text: experiment

text: presented

text: in

text: the

text: Section

text: it

text: is

text: natural

text: to

text: ask

text: analogous

text: question

text: for

text: the

text: case

text: of

text: points

text: sampled

text: from

text: a

text: d-dimensional

text: box

text: Theorems

text: on

text: the

text: random

text: Vietoris-Rips

text: and

text: Cˇech

text: complexes

text: can

text: be

text: found

text: in

text: The

text: results

text: presented

text: there

text: indicate

text: that

text: Betti

text: numbers

text: of

text: random

text: complexes

text: “do

text: not

text: live

text: together”;

text: see

text: Figure

text: in

text: Here

text: we

text: focus

text: not

text: on

text: the

text: Betti

text: numbers

text: and

text: their

text: limit

text: distribution,

text: but

text: on

text: a

text: task

text: of

text: dimension

text: detection

text: based

text: on

text: persistent

text: homology.

text: For

text: d

text: to

text: get

text: a

text: single

text: point

text: cloud,

text: we

text: sampled

text: points

text: from

text: by

text: sampling

text: each

text: coordinate

text: of

text: each

text: of

text: the

text: points

text: independently

text: from

text: an

text: interval

text: For

text: each

text: dimension

text: d

text: we

text: sampled

text: point

text: clouds.

text: As

text: in

text: the

text: previous

text: section,

text: to

text: refute

text: the

text: charge

text: that

text: our

text: detected

text: diﬀerences

text: are

text: only

text: due

text: to

text: the

text: increase

text: of

text: the

text: average

text: distance

text: between

text: points

text: in

text: higher

text: dimensions,

text: we

text: rescaled

text: the

text: boxes

text: so

text: that

text: the

text: average

text: distance

text: between

text: points

text: in

text: each

text: box

text: is

text: We

text: computed

text: persistent

text: homology

text: in

text: dimension

text: and

text: of

text: the

text: resulting

text: Rips

text: complexes.

text: We

text: obtained

text: persistence

text: diagrams

text: in

text: dimension

text: and

text: for

text: each

text: dimension

text: d

text: The

text: permutation

text: test

text: was

text: run

text: for

text: permutations

text: for

text: the

text: obtained

text: persistence

text: intervals

text: separately

text: in

text: dimension

text: and

text: in

text: dimension

text: In

text: all

text: cases,

text: it

text: never

text: happened

text: that

text: the

text: distance

text: between

text: shuﬄed

text: sets

text: was

text: greater

text: than

text: between

text: the

text: original

text: ones.

text: This

text: experiment,

text: together

text: with

text: the

text: one

text: presented

text: in

text: the

text: Section

text: show

text: that

text: low

text: dimensional

text: persistent

text: homology

text: can

text: be

text: a

text: good

text: tool

text: for

text: estimating

text: the

text: intrinsic

text: dimension

text: of

text: data

text: sets.

text: Knowing