Energydriven pattern formation
Formal Metadata
Title 
Energydriven pattern formation

Title of Series  
Number of Parts 
33

Author 

License 
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 

Release Date 
2006

Language 
English

Content Metadata
Subject Area  
Abstract 
Many physical systems can be modelled by nonconvex variational problems regularized by higherorder terms. Examples include martensitic phase transformation, micromagnetics, and the GinzburgLandau model of nucleation. We are interested in the singular limit, when the coefficient of the higherorder term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermallyactivated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds.

Keywords  action minimization AvilesGiga problem calculus of variations crosstie wall martensitic transformation micromagnetics microstructure 
Related Material
00:00
Area
Multiplication
Scaling (geometry)
Mathematical analysis
Price index
Applied mathematics
Cartesian coordinate system
Food energy
Perspective (visual)
Connected space
Mathematics
Computer animation
Physicist
Energy level
Partial differential equation
01:34
Axiom of choice
Statistical hypothesis testing
Randomization
Group action
Length
Multiplication sign
Wave equation
Insertion loss
Parameter (computer programming)
Mereology
Food energy
Dimensional analysis
Fraction (mathematics)
Manysorted logic
Velocity
Different (Kate Ryan album)
Negative number
Perimeter
Position operator
Moment (mathematics)
Physicalism
Determinism
Twodimensional space
Surface of revolution
Measurement
Curvature
Arithmetic mean
Parabolische Differentialgleichung
Calculus of variations
Phase transition
Normal (geometry)
Mathematician
Resultant
Spacetime
Point (geometry)
Interpolation
Functional (mathematics)
Real number
Inequality (mathematics)
Event horizon
Frequency
Mixture model
4 (number)
Inclusion map
Selfsimilarity
Kritischer Punkt <Mathematik>
Term (mathematics)
Interface (chemistry)
Boundary value problem
Differential equation
Addition
Distribution (mathematics)
Focus (optics)
Standard deviation
Scaling (geometry)
Weight
Forcing (mathematics)
Bound state
Algebraic structure
Volume (thermodynamics)
Line (geometry)
Sphere
Combinatory logic
Musical ensemble
Object (grammar)
Local ring
Gradient descent
11:29
Logical constant
Randomization
Presentation of a group
Equaliser (mathematics)
Correspondence (mathematics)
Range (statistics)
Water vapor
Mathematical structure
Fraction (mathematics)
Fluid statics
Mechanism design
Different (Kate Ryan album)
Negative number
Infinity
Funktionalanalysis
Lattice (order)
Category of being
Angle
Symmetry (physics)
Figurate number
Point (geometry)
Free group
Connectivity (graph theory)
Product (business)
Frequency
Inclusion map
Term (mathematics)
Divergence
Boundary value problem
Modulform
Nichtlineares Gleichungssystem
Standard deviation
Scaling (geometry)
Kälteerzeugung
Physical law
Algebraic structure
Counting
Volume (thermodynamics)
Total S.A.
Mortality rate
Line (geometry)
Cartesian coordinate system
Radius
Network topology
Entropy
INTEGRAL
State of matter
Length
Multiplication sign
Direction (geometry)
Parameter (computer programming)
Repeating decimal
Mereology
Food energy
Grothendieck topology
Order of magnitude
Mathematics
Positional notation
Manysorted logic
Cuboid
Circle
Körper <Algebra>
Series (mathematics)
Surface energy
Perimeter
Position operator
Area
Injektivität
Process (computing)
Concentric
Moment (mathematics)
Measurement
Discounts and allowances
Proof theory
Curvature
Vector space
Calculus of variations
Tower
Normal (geometry)
Right angle
Geometry
Functional (mathematics)
Interpolation
Inequality (mathematics)
Theory
Power (physics)
Centralizer and normalizer
Wellformed formula
Interface (chemistry)
Differential equation
Theory of everything
Hydraulic jump
Expression
Mathematical analysis
Evolute
Numerical analysis
Diameter
Film editing
Vector field
Calculation
Factory (trading post)
Integration by parts
Bellman equation
Vertex (graph theory)
Glattheit <Mathematik>
Object (grammar)
31:02
Axiom of choice
Group action
Equaliser (mathematics)
Range (statistics)
Dimensional analysis
Sign (mathematics)
Roundness (object)
Minimal surface
Saddle point
Conservation law
Convex set
Physical system
Theory of relativity
Constraint (mathematics)
Gradient
Maxima and minima
Category of being
Parabolische Differentialgleichung
Angle
Symmetry (physics)
Phase transition
Weißes Rauschen
Spacetime
Point (geometry)
Standard error
Sequel
Tournament (medieval)
Similarity (geometry)
Continuous function
Event horizon
Vortex
Frequency
Kritischer Punkt <Mathematik>
Hyperbolischer Raum
Term (mathematics)
Divergence
Boundary value problem
Modulform
Nichtlineares Gleichungssystem
Focus (optics)
Standard deviation
Scaling (geometry)
Physical law
Model theory
Mortality rate
Cartesian coordinate system
Limit (category theory)
Gradient descent
Entropy
Beta function
Length
INTEGRAL
State of matter
Multiplication sign
1 (number)
Insertion loss
Parameter (computer programming)
Mereology
Food energy
Manysorted logic
Thermal fluctuations
Square number
Integral equation
Series (mathematics)
Körper <Algebra>
Position operator
Perimeter
Area
Process (computing)
Sigmaalgebra
Generating set of a group
Moment (mathematics)
Time domain
Degree (graph theory)
Calculus of variations
Normal (geometry)
Right angle
Partial differential equation
Classical physics
Functional (mathematics)
Finitismus
Observational study
Inequality (mathematics)
Theory
Natural number
Operator (mathematics)
Interface (chemistry)
Differential equation
Hydraulic jump
Mathematical optimization
Condition number
Addition
Noise (electronics)
Evolute
Numerical analysis
Computer animation
Physicist
Calculation
Integration by parts
50:35
Group action
Dynamical system
Multiplication sign
Wave equation
1 (number)
Parameter (computer programming)
Mereology
Food energy
Dimensional analysis
Velocity
Different (Kate Ryan album)
Thermal fluctuations
Square number
Determinant
Perimeter
Position operator
Physical system
Area
Process (computing)
Theory of relativity
Moment (mathematics)
Physicalism
Maxima and minima
Lattice (order)
Thermodynamic equilibrium
Entire function
Curvature
Calculus of variations
Phase transition
Summierbarkeit
Resultant
Spacetime
Geometry
Point (geometry)
Standard error
Observational study
Event horizon
Theory
Product (business)
Kritischer Punkt <Mathematik>
Term (mathematics)
Profil (magazine)
Interface (chemistry)
Modulform
Energy level
Nichtlineares Gleichungssystem
Hydraulic jump
Initial value problem
Condition number
Addition
Focus (optics)
Chemical equation
Expression
Mathematical analysis
Bound state
Algebraic structure
Independence (probability theory)
Mortality rate
Line (geometry)
Limit (category theory)
Finite difference
Integration by parts
Musical ensemble
Gradient descent
00:00
The man let's get started our 1st speaker this
00:14
Warning and Robert call above got his Ph.D. from Princeton in 1979 and went directly to the prime Institute Where he's been ever since Seoul Korea he worked scenario nonlinear partial differential equations none of that area Problems always connected to significant application and especially in materials science Worries had tremendous impact is many celebrated contributions normally have a multi scale aspect revealing and surprising connections between behavior at the microscopic and macroscopic level His work is characterized by a broad perspective from the most concrete aspects of applications The deepest mathematical analysis is a superb example of an interdisciplinary mathematical scientist who could work effectively with physicists and engineers but always brings a mathematical way of thinking and analysis of the problems is held in very high regard by colleagues as indicated Manly Wade 1 of which is used not only a speaker at this International Congress would also be a plenary speaker at the International Congress of Industrial and Applied Mathematics next year Missouri my pleasure to introduce Bob Cone was big This morning on energy and that information
01:42
Thank
01:44
So I've been using this tidal energy driven pattern formation for a few years and that but is not a standard terms and peoples Frequently asked me what I did tonight Kept Turbo answering question I know it when I see it I think you'll know it when you see a band Part of the point of this talk is to help you recognize it and to do that by Talking about some exam so I intact My talk will be essentially to read thing Yet this as a couple of Corollary 1 is that if you get word or lost Be patient there will be a reboot and just few minutes and You'll see as we go along that there are some unifying themes All of my gets involved Nyman PT wore the captain's gradations or some combination of those they all come from physics and materials science and toward C and will try to capture some more teams that are in the 3 marks the 1st will be found on worsening raid the 2nd will be about microstructure crossed my wall The 3rd will be about Pathways of thermally activated switching And don't be concerned in each case I'm begin by telling you what the question before we talk about what's so let's start the 1st bounds and forcing read this as it is for that I did with A few years ago And this will see Interpolation inequalities and energy inequalities very familiar tools to people who work in peace will be at the heart of the matter so what's the quest unvented focus On the simplest problem of this type But I know I can do that involves So this is these pictures of their top Beazer numerical simulations to kind of worsening have talked about would normally in material science set in place in 3 dimensions But the question is The same into dimension so it's easy to visualize computer twodimensional so let's concentrate their We're talking about a twophase mixture of black blackandwhite and white interface that separates the 2 phases Is the object of interest hits We think it's India want talk about It is the 1 that says it's normal velocity is the tangible bosses of the curvature in the twodimensional setting The ordinary curvature velocity engine loss of net loss preserves the Long Beach face and just to keep things simple let's suppose the volume fractions are 50 per cent each and and Judy simulations were down with periodic boundary conditions on I'm really interested in the problems that feels so much space that the boundary effects are unimportant periodic boundary conditions are a convenient way to capture that America would be simulations were done with random initial data And the use of that That is meant to limit These distribution that's created by Let the French phase transformation and the common belief seen both in experiments and numerical simulations is that for random initial data There is an intrinsic like scale Well 1st an initial transient where the structure organizes itself a bit and gets to some stated that like the 1 on the left and then it evolves and as it evolves well with Features of the head and remain in much the same I hope you would be that these pictures all look pretty similar However the length scale increases and in fact The common belief is that The lights that Skilling pieces of our this is his most all look simple but if you try to write it down His People at Nonlinear 4th order Parabolic and Lesotho like skill Has any still still want time but also would have to be By emerging theater 1 4 And I believe is in addition that the solution is statistically self similar events Essentially the statement that Optus scaling up the the magnification that looks about the same time this particular revolution Is energy driven it could be thought of as steepest descent perimeter can be a suitable negative norm But I don't want a fast that so let me simply captured the consequence of the most immediate consequence of that And that that's The perimeter decreases along the well that's very elementary results based on the fact that curvature is the 1st variation of parameters OK so what This is an old question and visit more last nite even question is are Strongly held beliefs among star Is physics Valley and an obvious question is what's for mathematician to do now I think we all agree that We better be careful not to things that are wrong I don't play Only he said This I don't know what self similarity me whatever means I'm not quite sure it's true and The bottom line is I have nothing to to say about the motion self similar I am unable Describe these veterans in any detail And I am unable to explain to the sense in which there were statistically self what I want to tell you about is in a fresh approach To the scaling up to the idea that the lights Rose in a systematic way End We need origin of that really comes from asking the right questions this this statement that might still grows like you want for early Tuesday the 1st is that it never stopped worsening keeps on getting more worse all the time the 2nd is that it doesn't cost faster It doesn't get stuck Sorry that it doesn't suddenly disappear suddenly cost so fast that it reaches final size of the 1st in a statement that it doesn't get stuck here well also if I hear An arrangement of spherical inclusions for example of the curvature individually constantly lost the curvature would be 0 They wouldn't now that doesn't mean that random initial data gets stock but of what it means is That in order to prove the random initial they never get stuck you should have to to prove that it never beaches 1 of these critical points of the floor that's difficult because you have to somehow random I'm a determined determinist to guide So I'm kind quite concentrate time is set 2nd in a statement that the the solution cannot force faster and I tell you About I want those about proving version of that resources and 1 of the things to pick out here is that there are incredible Patents that former shooter physical phenomena and the others the well did we think carefully about it some part of what's happening can be approved maybe even with elementary means so that we start We were looking at a picture mathematically represent that by a function which is 1 in black and minus 1 and the White and into real test here I'm making a statement about the length scale as a function of time and I hope you're worried about the question of what exactly I mean by the scale That's really but hardly let's Keep our like simple by working in Korea tastes of the that's easy to talk about air quality and by sticking to people born fraction of and undertook waves The finding a sort of a maker scarf widescale scale 1 of them is the perimeter per unit volume flesh internal needs averaging and I'm going to talk about how these how we still have a moment this 1 will still like 1 over the local life skills by any reasonable Interpretation of local that's a positive Lebanon BV nor might call a different approaches to use a negative so that their different choices of negatives of the 1 I like to use for a convenient for these problems is the 1 defined here
11:37
It's Defined by duality by by taking the marker function and multiplying it against budgets function with Budget cuts unless the 1 so if you wish that the Doole tool W 1 infinity in the way our usual notation call them minus 1 1 and so sometimes just to remember where it sits in the scale of things I like to write it as if I were taking 1 integral of the marker function and they don't want all he L He is That's the energy firm in advance but think for a moment about why does Both each of those in its own way represents the like scale I told you already that I a circular or spherical inclusions they would so I don't expect to see it like that But still for understanding the scaling it's easy enough Easy to think about the fight In inclusion in a box of size of radius Diameter And if individual lights here so equal volume fractions of black and white The spacing between the circles is my little l let's natural still pictures are so And the marker function saying 1 in the black and minus 1 white tent certainly the perimeter promoted area using the count of the easy account area And that 1 over the local legs that and this negative sublet norm Well the interesting functions Jeter tried to be as large as possible work positive plus 1 trying be Smallest negative as possible where the marker function minus 1 but But these plus 1 regions in the minus 1 regions separated by scale of water L & G Which constant want so it can Really get much bigger Much smaller than mine itself so that product is going to be a border post a minus 1 plus or minus down so named by Mr. calculation you see that this scale that that this Negative norm skills like the still OK showed the mathematical Charlotte The subject is the fact that it well if we have 2 different ways of measuring the local would still should be related the products He Times that's like 1 of my clients like sort dimensionless and so it's a fact mathematical fact fact the functional analysis that debt Bounded below by announcer that's really just a special case interpolation inequality that interpolate array of urges Interpolate C L 1 between the BV norms and negative this kind of interpolation and followed isn't quite standards but not factory brood it I think I hope it would be reasonable because the right here Meets all the obvious scaling consideration and infect interpolation and like this to be used in a lot of energy and formation of essentially because in materials science Surface energy is often area or if you seem like the structures you can measure the bill by negative norm as we were just discussing and so somehow the 2 objects on the right recur again and again and applications from terrorists so far I haven't really said anything about the evolution of this evolution law Not only has the property the energy perimeter increases that DDT It also has the property that The LTTE Negative norm which is why measure of light scaled back its rate of change and time was related to the river and rate of change and that's in energy inequality the book is trivial you take equation multiplied by something and integrate 1st I'm not going so you prove we don't have time formulas like that but there's nothing to it a little intuition Why should the greater change perimeter control the rate of change of this negative norm Yes The lights quite changed and the interface must be moving and the interface must is moving because these motion is energy driven energy itself was because that's Jewish properties were basic PT Bank and so were left with very simple facts that we have 2 different ways of measuring the local still product is bounded below A couple of different Qualities And honestly when our 1st look at those sites as well so those probably shouldn't by scaling laws and surrogacy that's not quite true that as a matter of oldies forgive Campbell examples of that but it's true when so time Mayor Yuri perimeter per unit volume raised the appropriate 2 doesn't work with and there's a range of powers that worked out Is bounded below by the corresponding expressions with he replaced by the anticipated scaling so this is a statement that the From this very simple Only information we get As the time air bridge versions of the scale of the expected an area believed the trick is setting it up right once you realize how set up the proof is just an OBE argument more or less like from OK so that we achieved modest achievements are again these the pictures Show before capturing what Our capturing geometric evolution that I understand we still don't really understand what makes these evidence of characteristic and recognizable me I'll bet the injection that are statistically self similar but we've been able Prove for Brown on the forcing rate and prove that By an argument that has nothing about randomness and billion the trick was to set things right involving 2 different ways a positive and negative norm as ways of Measuring the local like scale and then to use interpolation and energy since fail it's an idea I introduced this technique and 2002 it's been used in a number of different problems by named a few years and a And by a number of people but I would say at the beginning it seemed like The best thing since sliced bread and on the other hand it turns out when set out to you is is that there are lots of problems motion by curvature of grain growth worsening of large numbers of plan foresees just The couple examples that I don't seem also easily took strict Even the question of how our particular problem can be set up These terms is not always That was the 1st example and A course that was dynamic and formation of those were Patterns that were forming in the course of evolution process which finally was heading For a very simple figures by 2nd example It is about the structure of across Taiwan Again I will tell you what is across time At the beginning this is mainly work that was done by a loose Riviera sir He in 2002 the viewpoint I'm gonna give you are is 1 that's very close to theirs Slightly different and presentation Developed with couple my colleagues and basing on Indian and As you will see the heart of the matter will be an optimal geometry independent over about which finally At the very bottom is an inspired integration what's across Taiwan so here we're talking about Magnetic thin films very thin Not to talk about magnetically soft material Permalloy example 1 in which an suffered is not important Solis It means the canyons of navigator and this picture is a little too small you're going to see a blowup of this picture of Obama but
21:20
Our were essentially you should have here will be capable of So will come back to that but But That's that's the pictures that you see as you look down at the top of the found magnetization what everybody's firmly with magnetization you have Magnets on refrigerators parliament and magnetization can be visualized by putting iron filings and filings lined up with that Local direction back position so this experimental technique that produce this picture areas More let's just no that magnetization is point gain in 1 direction onesided crossed out while the other direction the other said 1st of all in 1 1 looks closely at the region that black from the top of the tower and sees the veterans dependent on lap will talk about the parent in more detail The moment I'm going to try to explain to you why it happens that sort of interest rates the symmetry of the problem and I'm going to explain to you that it is an urgent of course in order to do that we have to talk about what is the editor what is a pattern In principle I wonder whether the energies of these parents and then to show something is minimal of course you have to prove that nothing else has Lowering OK crosstie and that's the better We were seeing how a little smaller before this is what you see as you watch the metal shavings lineup With B magnetic field looking at the top of that felt and Over here is a schematic of arms Let's concentrate on the man the the schematic magnetization vector Points east appear West lay down here this central line corresponds to the slick still lying here pictures and Here the magnetization is going along sir wolves in here years discontinuity direction along a vertical line discontinuity direction along line horizontally and there's some regions in the center where it is wise constant so what I've drawn here is Is hard to 1 of the pattern Which consists of Of peace What if Move back Fuel always magnitude hats Would use its Everywhere divergence Free even at this time that would ease At a discount Woody beam divergence free means the normal component is contained and so that's true even if these discontinuities here and here The normal on continue OK so indeed accusations in this problem the mechanization is independent of thickness of playing record field base West move divergence free back 1 and the puzzle the surprise you The geometry the geometry would have permitted It doesn't have a sharp discontinuity to grow sharply from east to west across the line but material chooses not to do that it chooses to go from east to west By developing new smooth pattern Which has discontinuities Internally along Well with vertical as well as far I know Question is what and I I would note said In drawing when you draw this They had a well if you That I'm never have circles or piecewise constant discontinuities along vertical horizontal lines shot and the district is completely determined except there's an internal scale the length skill with which it repeats for example and will come back to that at the end of so I think we've been over this already have forcibly defined magnetization had to be a divergence free be smooth vector field with magnitude and I will refer to the discontinuities Wall What's the energy of a pattern well There is a theory called electromagnetic at a more fundamental theory than what I'm sure hear it provides a scheme for calculating internal structure of these discontinuities and their energy and and wanted the time to take bets on just tell you the energy of the editor of the some of the the energies of the walls these smooth parts don't contribute energy at the meeting were and discontinuity socalled wall If if if the total while remember it's gotta be divergence trees The while angrily To fade As shown and the energy assigned data might stated cost per unit like so now we can sort of understand what's at fee If the energy from the length of The plan would be assigned data minus stated 0 signed For small angles that behaves like small Wang Moser very much cheaper than large and so this pattern Replaces 100 did we angle which would have been rather expenses by The Other smaller angles literature there were numerous but and in fact Doing a little calculus if you accept this as the geometry of all this His might tool of what's happening in Burma to accept this As the geometry of the wall you could calculate energy per unit horizontal lines and you get Where took might 1 were assigned data might mistake it 1 and so this perching certainly does better down the simple wool that you might have imagined OK But the real question was that the real question is why does the Magnetic and don't choose this Veteran in particular can simply calculating the energy of the Spanish dancer question answer that question that explained why no under that would have a look at the typical goal of this kind of static energy driven formation analysis of once we think we understand what's the energy and we think We see asserted pattern and would like to explain an explanation should beat geometry independent lower bound proof that no other would have lower and utterly do them off This is the part of what I wanna tell you up to now I just telling what's across wall The real question is how do we ever go by demonstrating that this pattern is off and Well answer this there's a few different ways of using different examples but many of them come down integration by parts and some clever form That's what I'm talking about So the scheme is a draw the region that is occupied by the trust I've always so so like her to miss something that would fit into here we imagine was boundary conditions The magnetization Above should pointed right far below which should point to the left and and what I want to allow some kind of carried this city's so I will do that by imposing periodic boundary conditions on the side without assuming anything about the like period and the scheme proving lower Brown series introduced What I like to call and entropy Area vectors functions of the magnetization Vector value function of the magnetization which has eased He track the 1st is that when Will divergence free unit that vector field there this city of vector field also divergence and the 2nd is that whenever job Across a discontinuity a divergence You with have will say that the jump simmer down The normal Is related to my energies signed a state of books and in the important point is that if whenever you come up with record field like this that a lower back because for any candidate any divergence free unit magnitude peace was effective field we could so that the heart of this between this inequality and that this 1 we have Green's formula the integral of
31:24
The normal Sigma died and the boundary of my rectangular domain is Really is the integral jury divergence of in equality what I have a value on the lefthand side that determined by the amount of data on the right hand side well the divergence of stigma is concentrated on the walls at the walls of the divergence is the jump in signal about normal and so that's actually Western Wall energy so this righthanded but this is this Completely elementary think Why do I call center those of you who were familiar with at Kabala conservation laws will recognize that You will know that whenever you have a hyperbolic conservation laws Dealers At the Bala conservation law once based I and generate additional conservation laws we call entropy People spirit have their conservation laws that were formerly implied by changing when you started with can Here to give very 0 Louis while I'm in the minds of that's like racial laws and didn't single Obama equals 0 so similar members like 1 of those derived conservation so Mrs. you was like an entropy and clubs now There are a lot of these Most Justice and still consummation a lot mentor for people bears but of course We need a special need 1 which will prove that The pattern that we see is really out sold you look around a little bit And the 1st thing you think you're writing down as this 1 I wrote down here if and EDI fado Sigma them It is Bayer and plus and rotated by 90 degrees so that would be data so simple 1 would be data cosigned a minus sign and playing calculation checks that This Choice So much as the desired Property of being a derived conservation when in this movie that's just The 2nd condition that John B. and save if and Discontinuous But divergence read Across the wall Then the job and segment . who is related to my while energy is is not only true accords with equality again that's just just a simple calculation and if this relation holds with quality and we would expect This relation with a quality Is 1 with equality The all ones side so that has a pretty good chance of giving equality But depending on the Riza problem the problem is that the patterns I'm thinking about The anger was not well defined on being was well defined and missed our but But Their vortices there Each of these little points here for example assembled around just fight here magnetization on on the left that's going on Friday points out you start to think about it winding degree 1 round there so Inside the walls There are 4 season there for the anger was not welldefined OK that's Actually not a technical point A very important point but I want to get around at the EU choice that I wrote down can be reflected that you take that choice Essentially translated in 1 quadrant and then reflected suitable you get something all or don't don't worry about the form the important point is that 1 can find a continuous function which is welldefined on does not require a choice of Beta taking values to tool and which which has all the properties we talk about war and which agrees with my 1st in 1 so that it achieves equality rather than just inequality gift from all angles less so now you can see that Effects Occurred to me that I drew achieves equality in my lower bound and I don't have to do any integrals to see it I just have to observe that the walls All Less than I in fact any pan out which restricts itself was less than 90 degrees would achieve the optimal and now I did mention that there is an internal length scale so again this So this is out of all of the evidence optimal because the wall angles Rather than being a 180 degrees As example for all restricted the last half what sets the internal lights Well let's set by things that I've been ignoring my discussion and just Mentioned the film's experiment experimental films usually gets and some and Isak Rapidan this Wallace here in the 1st place the magnetization refers to be horizontally very pleased wet and somehow the top regional and eastern about region went West And in the middle they have to somehow find some way of accommodating the strange but in addition Win The thickness It is relatively large relative to socalled exchange ratio Parameter in the magnetic model and these walls have long tails they tend tell each other so and I suffered before prefers the small The loss of El each other which 1st internal The large and that's it So wrapping up the crossed by Wall is an interesting patterns in part because it's so characteristic receipt is and every time you look at magnetic thin films in a certain range effect it's interesting because it's sort of breaks the symmetry of that set by me from the boundary Commission are achievement was to blame Well It's there because no other better and have lower than the truth finally Barnum was nothing but inspired integration by parts and I'd just like to to mention that similarly inspired integration by parts of Many areas of the couple Variations in Study minimal surfaces Means calibrations prove that surfaces The normal that's really Similar integration by parts in some areas of energy grid pattern formation yesterday so All relaxed energies that went on for example in martensitic phase transformations and there in a Lagrangian interviews but they play very similar I am an obvious question is whether it is the type of argument I described using entropy using integration by parts was here luck or whether Problems like this Can always be treated by such arguments and well I think I don't know the answer the 3rd exam 3rd example There's more recent work on pathways of thermally activated switching here we will again be doing a variational problem But We will be doing it in the setting a fine and Mrs. Thyssen work that I've been doing with Allen's listed here so the focus is on well a physicist who called this the planned out and burp functional And calculus variations that often goes by by the name a modicum mortal of functional if you heard of did you Talk a couple of days ago He was talking about critical points of the functional if I concentrate on Convex domains or periodic boundary conditions and the only local minima are the obvious ones where you identically plus or minus 1 I'm interested the situation were excellent small also you who prefers to be infrastructure 1st 4 B plus minus 1
40:51
If it changes from 1 to the other there will be a strong contribution of this term which will essentially out of higher dimensions the perimeter of the interface and in 1 dimension the number of interface question I 1 addresses whether the pathways of thermally activated switch Notice that all my said Pictures and this is a picture that come with that But I have 2 questions before It's capitalist variations we often spend a lot of time studying global memory but of course we're interested in local minima and nature is especially interested and everybody knows that major finds minimize or glass of beer you get that nice fall on top it looks like it's there forever and you come back 5 minutes later and got are system use physical systems escape from local minima in various ways But the simplest example of such away from fluctuations and energy driven system a system whose evolution would be easy is minus radiant some energy The natural way to model Thermal fluctuations noise and make this well that's a finite dimensional Odie then it becomes a stick Which is the noise will I always take you out of local minima And some other neighboring minimum but of the noise is small Then they escape his mayor takes a long time now we're eventually important Anybody who bought a computer and had their hard drive failed 2 weeks later knows that Rare events happen And tho they may be reared they may be the things that trouble was not so we will be studying in a finite dimensional said Suppose our system is just being a busy month he complained and energy A double well symmetric energy prefers to points on the 1 axis and Mr. Kessel differential equations would be what I wrote down here are in the car to just not a very good character because it would want much smaller than my car to before finally get finds its way up The seperate takes over Well very nearly all the saddle and then goes down to the the other large deviations principle Tells us how transitions will take place And Probabilistic speaking When it will take place in the limit of small and a simple statement of that Is that if you assume that the transition takes place with a certain fixed amount of time and if you and the limit of small noise and an estimated probability of that transition of the event text Small but more than that once the treasure takes place The 1st with very high probability by the path that minimizes this tournament operational problems this is called the action it's really just equation In this example might equation Germanist operation without was Z that miners Greek so C . plus waiting empty square Equation that's a mountain noise that we want to make the transition pocket So Just a little more in terms of lesson about Action minimization And the impact of thermal fluctuations and if we permit that So we're talking Barbara events they usually take a long time have done in the limit Switching that takes a long time It's very easy to describe the minimize sir This variation from this action in position major problem while minimizing at the path that minimizes equation error integrated from 0 to the final time goes up field From the critical point to start To the lowest round and then go straight down and easy proved so let's take a moment to do that if I take any intermediate time in integrated equation error from 0 times in now Just elementary manipulation tells me that they had Z T minus payTV squared plus The cross her Christer integrate by parts so it really e and my kind out whatever My Iraq the other terms negative solidly vacationers greeted the sequel to the energy yet Any intermediate time minus energy times around the natural choice now is the time when you cross The bridge A separate the 2 and And so the shows that using their choice of how to minimize the hand side like to minimize hand side which means they should cross the bridge point by way of dissent That's why Classical nucleation theories all about settled his argument breaks down when the time remaining before switching Large because in order for the stars in order for me to do that argument says it goes the saddle I must Be able argued that the other turned away can be made negligible and that's only true when capital has so I look at the rarest events for which the switching time effects The optimal halfway doesn't need to to go through a saddle but as we said Events even very rare events are still interested OK let's get back to the book Can the Landau aboard A lot of this energy that I set out initial double well symmetric potential perturbed by gradient with a small perhaps steepest descent is now Parabolic differential equations goes by the name of the young contemplation if I perturbed by Norway's that's stochastically perturbed Partial differential equations richer theory of event 1 space to mention the series space to mentions is not very complete and it's hard to interpret the solutions by insisted on using Browning motion white noise in space as by perturbation understood PC gave a talk Couple days ago which begins to address whether that's quite the the right idea from the viewpoint of action minimization 1 of the advantages of excellent position is we don't have to worry about the action minimization is determined operational problems and I would submit That any reasonable A reasonable interpretation of the noise return version of this problem should permit treatment by action and so we go straight So steepest descent as I said As well At 1st would say she was minus great he what's the gradient well that's the 1st creation so It was the 1st variation operator time as excellent those 2 0 the time scale of motion also changes And the time scale are in order to keep the timescale emotional of border 1 we should work with me skills that The correct Allen contemplation and actually minimization man years equation error integral equation Hereford this suitably be scaled the result be of what skilled version of T plus a skilled pollution square In our interests Is an understanding What happens when you minimize this subject to the constraints they use identically minus 1 initial I'm an identically plus 1 at the final climb and focusing on the sharp interface So let me just tell you Bianchi in 1 space dimension let's use periodic boundary conditions to be very specific and that's what I'm tell you was completely broke off The optimal pathways Mysteries describe it knew sold start from being identical minus 1 and then I'll immediately nucleate in seats Equally spaced periodically arranging equal space and A seed In 1 space to mention 2 sides so and And wolves and then member action minimization This is an integral in time as well as they His as it proceeds in time describing pathway they get from the uniform minus 1 state for us want state and after the creation of an all
50:39
All that happens is the seeds expanded Velocity until the roles and to into space to mention use the pictures Well with similar but little different I wanna be clearer that mentions it's really Formal rather than the Call this conjecture that I you expect respect the creation of so so again into space to mention to begin identifying minus 1 selects White Minus 1 and then you will create seeds so black and traded seats 1st difference until dimensions is that If the seeds nucleate as points than the perimeters practically 0 so someone 1 space to mention of additional costs Energy immediately to space to mention of half the seeds will expand eventually emerge and eventually eventually takeover Completely until everything's all black so another difference is that I am not so what we expect we expect that In the limit Sharp interfaces From our action and musician problem should be the minimum minimization over after waves switching efforts of half of the cost of nucleation which could be easier the based pointer patients has passed and the other terms would account for the motion and here's what we expected The velocity curvature squared wise that well Because you expect equation error for The limit of violent calm Malone the male encounters motion curvature of this This would make the system want go Appeal Backward motion by curvature and then down filled the emotion prove true if it has enough time to do some some comments about about that picture But I'm calling nucleation events These are Thermal Aren't the thermally activated switching but they have nothing to do with Critical points or saddle points of the functional so this is The type of nucleation that's very different from Classical fishing I told you that the onedimensional treatment is completely rigorous And 1 and think it through during much but Of course get started the 1st thing you have to to do is to prove that Those 2 0 you're talking about Funny points which removing in a continuous fashion and then so it had to do a structure for the limit before it started but the comment I wanna make said Finally part of a new era entire Reidy of studying thermally activated events and the sharp interface limit but what I'm talking about is closely related to censure interface limits of steepest descent l found British and the fact that they bad But the limit to ban motion Overtures toward more space to mention This is something that is sorted out about 10 years ago a very beautiful Theory there's a more recent aspects were also Talking about something very close to ejection George recently settled by sold there were some early results by others are really Georgia's conjecture involves the sharp interface limit of radio Community 1st creation square So Closely related and talk so well I don't want me to work too But I'm feeling that I should give you a little hint of what's underneath the analysis So let's just at the moment rebound talking about action minimization action in the position finally is the time role of this expression and became 1 of the ideas of the heart of this study is that The action controls the energy minimizing their action energy level 3 large and reasonably well From there jump in energy and the creation of an example that makes the energy job will cost action and that's Just the same argument that showed before a fight integrate our reaction in time From before The nucleation event right after placing name before replaced by U T minus Corinthians plus across and the 1st term is positive and the 2nd term integrate by parts to the difference in the images of the soul 50 Energy jumped up some so that was the end by reaction controls wall prop Profiles of velocity same argument but now going from initial time of the final time if I go from the initial time for the final time my initial conditions my final conditions are use and and plus lines and minus 1 this term vanishes So By then argument cross terms if I am going from 0 final and across Ginevra intrastate and just 1 Radick turned show up and I get to turn duties square and geese scaled back problems Now you can see Why do Georgia's conjecture The closely related OK the 3rd point I think I'll spare you that and that's just ran solo in this 3rd being yet The physical idea was focus on thermally activated switches and fixed time were studying extremely rare event but sometimes those are the ones of the most interesting on the mathematical side the idea is to think over the sharp interface of action minimization as in the past Show interface problems in the area I told little bit about the solutions from Ginsburg went down Monday in some form or analysis from Innsbruck Landau and to lead an obvious question is whether something like this can be done for physically more Complex and relevant problems OK but wrap up whole talk so I've given you 3 example promise to sums were Attempt to define energy Driven ever formation through examples 3 examples were Bounds and worsening greater across Internal structure thermally activated switching they were different in character and worsening rates we were talking about patterns that form and changing I'm so energy formation that dynamic thinker Frost wall structure we would talk about patterns that are really calculus of variations problem energy minima global minima and Geometry and balanced Explain why the patterns and cheat global thermally activated switching we were talking about well physically thermal fluctuations To cast a partial different Operations but Practically speaking to determine history Couple relations problem represented by action which users certainly not the only you Examples of energy compare formation even this meeting as many examples of energy during formation amongst examples that error Similar to the ones that I described character coming from the physical sciences Recent work A lot of recent work has been done on 20 of the martensitic phase transformation demand patterns are magnets for text take superconductors Soviet 50 book about yesterday now common well I'm and all these questions about the origins in the physical sciences the answers Offering analysis of an offering is involved speed ease an calculus variations the Roland G. driven but in different ways So we now it equilibrium and Well in this area there I think it's too early to try to define energy independence formation too willing to try to have universal I think the right Approach now for many years probably still will be to focus on not just at the end of the product names Clothes I collaborators again in each area and some people very closely related work And much of the work speaking about his been on my own and that has been funded by himself 1 feature acknowledged thank her