PHYSICAL REVIEW E VOLUME 60, NUMBER 2 AUGUST 1999 Functional integration approach to hysteresis G. Bertotti,1 I. D. Mayergoyz,2 V. Basso,1 and A. Magni1 1Istituto Elettrotecnico Nazionale Galileo Ferraris, Corso Massimo d'Azeglio 42, I-10125 Torino, Italy 2Electrical and Computer Engineering Department, University of Maryland, College Park, Maryland 20742 Received 22 February 1999 A general formulation of scalar hysteresis is proposed. This formulation is based on two steps. First, a generating function g(x) is associated with an individual system, and a hysteresis evolution operator is defined by an appropriate envelope construction applied to g(x), inspired by the overdamped dynamics of systems evolving in multistable free-energy landscapes. Second, the average hysteresis response of an ensemble of such systems is expressed as a functional integral over the space G of all admissible generating functions, under the assumption that an appropriate measure has been introduced in G. The consequences of the formulation are analyzed in detail in the case where the measure is generated by a continuous, Markovian stochastic process. The calculation of the hysteresis properties of the ensemble is reduced to the solution of the level-crossing problem for the stochastic process. In particular, it is shown that, when the process is translationally invariant homogeneous , the ensuing hysteresis properties can be exactly described by the Preisach model of hysteresis, and the associated Preisach distribution is expressed in closed analytic form in terms of the drift and diffusion parameters of the Markovian process. Possible applications of the formulation are suggested, concerning the interpretation of magnetic hysteresis due to domain wall motion in quenched-in disorder and the interpretation of critical state models of superconducting hysteresis. S1063-651X 99 06308-4 PACS number s : 02.50.Ga, 75.60.Ej, 05.40. a I. INTRODUCTION applying to increasing and decreasing H, respectively. From the physical viewpoint, this construction amounts to assum- The study of hysteresis has been a challenge to physicists ing that the system, once made unstable by the action of the and mathematicians for a long time. In physics, hysteresis external field, jumps to the nearest available energy mini- brings all the conceptual difficulties of out-of-equilibrium mum, which means that one excludes the presence of inertial thermodynamics 1­5 , first of all the fact that we do not effects, which could aid the system to reach more distant know the general principles controlling the balance between minima. stored and dissipated energy in hysteretic transformations The method discussed in Sec. II translates this picture into 6,7 . In mathematics, on the other hand, the central issue is a well-defined mathematical formulation, based on the fol- the formulation of sufficiently general mathematical descrip- lowing two steps. tions grasping the essence of hysteresis beyond the limited i Given the time-dependent input ht and the continuous interest of ad hoc models 8­11 . function g(x), analogous to the free-energy gradient F/ X In this paper, we introduce and discuss a formulation of hysteresis of some generality, inspired by the following situ- ation, often encountered in physical systems. We know that in physics, hysteresis is the consequence of the existence of multiple metastable states in the system free energy F(X) the temperature dependence is tacitly understood , and of the fact that the system may be trapped in individual meta- stable states for long times. Let us consider the simple case where the state variable X is a scalar quantity and the rel- evant free energy in the presence of the external field H is G(X;H) F(X) HX. The metastable states available to the system are represented by G minima with respect to X, for which G/ X 0, 2G/ X2 0. When H is changed over time, the number and the properties of these minima are modified by the variation of the term HX. The conse- quence is that previously stable states are made unstable by the field action and the system moves to other metastable states through a sequence of Barkhausen jumps. Because the condition G/ X 0 is equivalent to H F/ X, one can analyze the problem by using the field representation shown FIG. 1. Free energy F(X) with multiple minima and corre- in Fig. 1. The response of the system, expressed in terms of sponding gradient F/ X. The dashed line represents the hysteretic H(X), is obtained by traversing the upper and lower enve- behavior of H(X) obtained from the stability condition H lopes to F/ X shown in the figure, the former and the latter F/ X. 1063-651X/99/60 2 /1428 13 /$15.00 PRE 60 1428 © 1999 The American Physical Society PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1429 of Fig. 1, one associates with them a certain evolution operator T g (ht), which expresses in mathematical terms the envelope construction of Fig. 1 Sec. II A . The function g will be called the generating function of T g (ht). The evolution operator acts on a given initial state s0 associ- ated with the initial input h0 and transforms it into the final state st T g (ht)s0 . ii Let G represent the functional space of all admissible generating functions. Then one constructs a general hyster- esis operator as the parallel connection of the collection of operators T g (ht) obtained by varying g over G, with appro- priate weights described by some measure on G. Hence, one arrives at the following formulation, in which the overall state St describing the collection is expressed in terms of the functional integral, FIG. 2. Generating function g(x) with illustration of the enve- lope construction broken line associated with the function h g (x;x0) and its inverse x g (h;x0). St T g ht s0 d g , 1 G II. MATHEMATICAL DESCRIPTION OF THE MODEL In this section, we will discuss in more detail the various where s0 may depend itself on g Sec. II B . ingredients defining the structure of the model: input histo- The generality of the formulation comes from the general ries, generating functions, admissible states, input-output re- nature of the space G as well as from general ways of as- lationships, stability properties, and finally the functional in- signing a measure on this space. We will show that several tegration over G. known mathematical descriptions of hysteresis, like the Prei- sach model 9,12 , are particular cases of Eq. 1 , and we will discuss some new connections that emerge from the A. Hysteresis in individual systems broader perspective offered by the functional integral formu- Let us consider an individual system characterized by a lation. A case of particular interest to physics is when Eq. 1 particular function g(x). The system is acted on by the scalar is interpreted as the average hysteresis response of a statisti- time-dependent input h cal ensemble of independent systems, each evolving in a t and generates the scalar output xt in a way dependent on the function g(x). different free-energy landscape. The space G acts then as a (a) Input histories. We shall consider input histories h(t), probability space and the measure describes the probabil- t 0, such that, at any time, h ity that an individual system of the ensemble is characterized L h(t) hU , where hL and hU are fixed given fields, delimiting the input range of interest. by a particular generating function g G. In this case, there They will be termed lower and upper saturation field, re- are situations that can be analytically investigated to a con- spectively. The function h(t) will be assumed to be piece- siderable degree of detail, first of all the one where the gen- wise monotone. erating functions g(x) are interpreted as sample functions of (b) Generating functions. Let us consider a given output a continuous Markovian stochastic process Sec. III . In par- interval x ticular, we will show that homogeneous processes give rise L ,xU . The function g(x) is an admissible gener- ating function associated with the interval x to Preisach-type hysteresis, and we will derive explicit ana- L ,xU if it sat- isfies the following properties see Fig. 2 : lytical expressions for the Preisach distribution as a function of the parameters governing the statistics of the Markovian i gis continuous in xL ,xU ; process. The results obtained in this paper can be important in applications to physics, where randomness due to structural ii g xL hL ,g xU hU ; 2 disorder often plays a key role in the appearance of hyster- esis effects. The equivalence between Markovian disorder iii h and Preisach-type hysteresis implies that the average system L g x hU for any x in the range xL x xU . response under small fields is parabolic, a result well known in magnetism under the name of Rayleigh law 12 . In su- (c) States. Any ordered input-output pair s (h,x) is an perconducting hysteresis 13 , the same equivalence might admissible state for the system. We will be mainly interested be of help in the interpretation of critical state models in equilibrium states, defined as the states of the form s 14,15 in terms of the statistics of the pinning sources acting g(x),x... with xL x xU . In other words, an equilibrium on Abrikosov vortices, given the equivalence between this state is represented by a point on the generating function. All class of models and the Preisach model 9,16,17 . Con- other states will be generically termed jump states. Given the versely, a limitation of our formulation is the fact that it is generating function g(x), an equilibrium state is fully de- based on independent single-degree-of-freedom subsystems, scribed by its output x. In this sense, we will often identify and is thus expected to yield an incomplete description of an equilibrium state h g(x),x... simply by its x value. hysteresis effects arising in systems with more complex in- When the input h is given, the possible equilibrium states ternal structures 7,18,19 . under that input are obtained by solving the equation 1430 G. BERTOTTI, I. D. MAYERGOYZ, V. BASSO, AND A. MAGNI PRE 60 st ht ,xt T g ht T g hn ŻT g h1 s0 . 5 In particular, the output xt can be expressed in the form xt x g ht ;xn hn ,...,h1 ;x0 ..., 6 where xn-the output value at the last reversal point- depends in general on all past reversal inputs. Note that the evolution is rate-independent, because the state st depends only on the current value of the input and on the sequence of past reversal inputs, regardless of the input rate of change. A relevant aspect of the evolution described by Eq. 5 is that it exhibits return-point memory also called wiping-out property 19­21 . By this we mean that, given the initial equilibrium state s0 (h0 ,x0) and the input extrema se- quence h0 ,h1 ,h2 ,h1 , with h1 h0,h0 h2 h1 , then FIG. 3. Action of evolution operators T g (h) associated with a T g (h1)T g (h2)T g (h1)s0 T g (h1)s0 identical conclu- given sequence of input reversals. The initial state is lower satura- sions apply to the case h1 h0 , h0 h2 h1). In other tion sL (hL ,xL). Reversals take place at s1 T g (h1)sL words, when the input returns back to the first reversal value h1 ,x g (h1 ;xL)..., s2 T g (h2)s1 h2 ,x g (h2 ;x1)..., and so on. h1 , the system returns back to the exact same state it occu- pied when the input first reached the value h g(x) h. In general, more than one solution will exist. Only 1 , and the effect of the intermediate input extrema is wiped out. To prove the the states sL (hL ,xL) and sU (hU ,xU) are unique by defi- existence of return-point memory, we begin by remarking nition. They will be termed lower and upper saturation state, that return-point memory is a property of any system whose respectively. We will assume that the state of the system time evolution satisfies the following properties 19,12 : i before any action is made on it is always an equilibrium the evolution is rate-independent; ii there exists a partial state. ordering relation among the states of the system; iii order- (d) Auxiliary functions. Given the generating function ing is preserved during the evolution of the system under the g(x) and the equilibrium state x0 , let us introduce the func- action of ordered input histories. tion h g (x;x0), defined as see Fig. 2 Property i is the direct consequence of the definition of the evolution operator T min g if x g (h). For what concerns property L x x0 ii there exists a natural ordering relation deriving from the x,x h 0 fact that an equilibrium state is identified by its output value. g x;x0 max g if x , 3 0 x xU In fact, given the equilibrium states s1 (h1 ,x1) and s2 x0 ,x (h2 ,x2), we can simply state that s1 s2 if x1 x2 in the usual sense. Finally, property iii is the consequence of the where the symbols ``min'' and ``max'' indicate the mini- theorems of Appendix B, which show that the ordering just mum and the maximum of g(x) in the specified interval. defined is preserved by Eq. 5 under the application of or- Function h g has the character of a nondecreasing envelope dered input histories. Therefore, return-point memory is in- to g(x), more precisely, of an upper envelope for x x0 and deed a property of Eq. 5 . a lower envelope for x x0 . The inverse of h g (x;x0) will As discussed in 9 , p. 13 , return-point memory has the be denoted by x g (h;x0). The mathematical aspects of the consequence that the final state s connection between h t defined by Eq. 5 is con- g and x g are discussed in Appendix trolled assuming, for simplicity, that the system is initially A. in the lower saturation state by the alternating sequence of (e) System evolution. Let us introduce the following evo- dominant extrema h lution operator T M1 ,hm1 ,hM2 ,hm2 ,... contained in the g (h), defined in terms of the function full reversal sequence h x 1 ,h2 ,...,hn . By this we mean that g (h;x0): given the equilibrium state s0 (h0 ,x0), with h h M1 is the global input maximum in the time interval 0,t , 0 g(x0), and the input value h, the state s obtained by h applying the input h to s m1 is the global input minimum in the time interval tM1 ,t , 0 is given by the expression where tM1 is the time at which hM1 is reached, and so on. (f) Stability properties. They can be conveniently de- s T g h s0 h,x g h;x0 .... 4 scribed by introducing the concepts of strong and weak sta- bility. Given the equilibrium states sA (hA ,xA) and sB The evolution of the system is constructed by applying (hB ,xB), with xA xB , we will say that xA and xB belong T g (h) many times in sequence, once for each input rever- to the same strongly stable interval if hA hB and sal, as shown in Fig. 3. More precisely, let us suppose that at T g (h)sA T g (h)sB for any input h. This definition gener- the initial time t 0 the system is in the equilibrium state ates a partition of the interval xL ,xU into disjoint strongly s0 (h0 ,x0) and let us apply the piecewise monotone input stable subintervals, separated by unstable parts. Conversely, history h(t). Let us denote by h1 ,h2 ,...,hn the sequence of given the equilibrium states sA (hA ,xA) and sB (hB ,xB), input values at which the input is reversed in the time inter- with xA xB and hA hB we will say that the segment val 0,t , and finally let ht be the current input at the time t. xA ,xB is weakly stable if g(xA) hA , g(xB) hB , and Then, the state st of the system at the same time is given by hA g(x) hB for any x in the range xA x xB . Notice that PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1431 ensemble to the common input ht and, roughly speaking, we sum up their responses. The formalism whereby we will carry out this sum in precise mathematical terms is the fol- lowing. Let us suppose that the response of each individual system is described by some quantity q g , dependent on the generating function g. The ensemble value of that same quantity, say Q we will use capital letters to denote en- semble properties , will be expressed as a functional integral of the form Q q g d g . 7 G FIG. 4. Illustration of intervals with different kinds of stability. Equation 7 is to be interpreted in the following way. The AB, weakly stable; A B , strongly stable; CD, weakly unstable; symbol G denotes the functional space of all generating C D , strongly unstable. functions satisfying Eq. 2 for some xL and xU . In general, x the interval L and xU will be different for each g G. One introduces a xL ,xU is always weakly stable by definition. convenient set a so-called algebra of subsets A G and a When the equilibrium states sA (hA ,xA) and sB (hB ,xB) positive measure defined over that algebra, (A) 0. Then are such that xA xB and hA hB , we will call the segment one assumes that there exist elements of the algebra giving xA ,xB weakly unstable if g(xA) hA , g(xB) hB , and rise to values of q h g inside any arbitrarily small neighbor- A g(x) hB for any x in the range xA x xB . A weakly hood of a given value q unstable interval will contain in general some strongly stable g x, and uses the measure of these subsets to calculate the Lebesgue integral of q subintervals. If it contains none, it will be termed strongly g over G, represented by Eq. 7 . To make this loose description math- unstable. The various possibilities are shown in Fig. 4. One ematically rigorous, one should resort to the language and can verify from Fig. 3 that, given any two subsequent rever- the methods of measure theory 23,24 . However, it is not sal points (hk ,xk) and (hk 1 ,xk 1) associated with a certain the purpose of this paper to go deeper into these mathemati- input history, the output interval xk ,xk 1 is always weakly cal aspects. In the following analysis, it will be sufficient to stable. In a sense, the evolution operator T g (h) provides a assume that Eq. 7 does have a precise meaning as a func- mechanism to select the weakly stable portions of the given tional integral, and that one knows how to assign the mea- generating function. This feature will play an important role sure in specific cases. In Sec. III, we will discuss a par- in the general formulation of Sec. II B and in the particular ticular case where one explicitly constructs the measure cases discussed in Sec. III. and expresses the result of the functional integration in a Stability considerations are important, because the evolu- closed analytic form. tion of the system under varying h is reversible inside each As a first step, let us apply Eq. 7 to the definition of strongly stable subinterval, so that its hysteresis properties ensemble equilibrium states. The main difference with re- are essentially governed by the sequence of jumps occurring spect to Sec. II A is that we can no longer identify an equi- from one stable subinterval to another. A system initially librium state by its output value. In fact, given the individual occupying a state inside a weakly unstable interval will never output x, the corresponding input g(x) may not exist for be able to come back to this interval if it ever leaves it. certain g functions if x is outside the function domain Therefore, only the strongly stable subintervals that do not x belong to any weakly unstable portion of L ,xU ), or may be different from function to function, xL ,xU control the which is not compatible with the assumption that the entire permanent hysteresis properties of the system. Two generat- ensemble is driven by a common input history. In fact, in ing functions possessing the same set of weakly unstable order to construct a meaningful equilibrium state, we must subintervals and differing only in their values inside these i specify the input value h intervals will give rise to identical hysteresis properties. Con- 0 ; ii determine, for each gen- erating function g G, the set of solutions of the equation siderations of this kind permit one to recognize certain quali- g(x) h tative aspects of hysteresis independent of the details of g. 0 ; iii for each g, select one of these solutions, say For example, Preisach-type hysteresis, briefly discussed in g (h0), according to some rule, and build the state s the next subsection, arises from generating functions con- g (h0) h0 , g (h0)...; iv construct the ensemble state S0 as taining two strongly stable intervals separated by a weakly unstable part 22 . S0 s g h0 d g h0, g h0 d g . 8 G G B. Hysteresis in system ensembles Let us now consider an ensemble of systems of the type Equation 8 shows that a great number of possible equilib- discussed in the preceding subsection. Each system is iden- rium states are associated with a given input h0 , as a conse- tified by a particular generating function g(x) whose domain quence of the various possible choices for g (h0). Only the xL ,xU will be in general different from system to system. lower and upper saturation states are unique, because the We wish to investigate the global hysteresis properties that equations g(x) hL and g(x) hU admit just one solution, we obtain when we subject the individual systems of the xL and xU, for each g. 1432 G. BERTOTTI, I. D. MAYERGOYZ, V. BASSO, AND A. MAGNI PRE 60 The ensemble evolution is obtained by applying Eq. 7 to Eqs. 4 ­ 6 , that is, S T g h s g h0 d g G h, x g h; g h0 ...d g , 9 G St ht ,Xt T g ht T g hn ŻT g h1 s g h0 d g , G 10 FIG. 5. Typical generating function associated with the Preisach Xt x g ht ;xn hn ,...,h1; g h0 ... d g . 11 model solid line , and corresponding envelope construction G dashed line, see also Fig. 3 . The dotted lines give examples of different weakly unstable behaviors giving rise to the same hyster- Because return-point memory is a property of each indi- esis properties. vidual system, it will also be a property of the ensemble evolution. The interest of Eq. 13 lies in the fact, discussed in Sec. III, The formulation summarized by Eqs. 9 ­ 11 is rather that the probability density p can be explicitly calculated in general and powerful, but it is also quite abstract. It is not the case where the measure is generated by a continuous obvious how one could possibly determine the measure Markovian stochastic process. associated with particular cases and carry out the functional We conclude this section by showing, as an example, integrals. In this connection, a situation of interest is when when Eqs. 9 ­ 11 can contain and reproduce other known one is dealing with a statistical ensemble of independent sys- hysteresis models. We will discuss the Preisach model tems, and one wishes to calculate statistical averages over the 9,12 . To this end, let us consider the case where the integral ensemble. In that case, Eq. 7 translates into mathematical of Eq. 7 is restricted to the subspace GP G containing the terms the physical idea that Q represents the sum of all the generating functions of the type shown in Fig. 5. The domain individual contributions q x g , each weighed by its probabil- L ,xU is equal to 1, 1 for all functions. Each function ity d (g) to occur. Accordingly, G must be endowed with is made up of two strongly stable, vertical branches 25 , the structure of a probability space: the elements A of the separated by a central, weakly unstable interval. The left algebra represent the admissible events that may occur in branch increases from h hL to h a at x xL 1, and the experiments, the measure satisfies the postulates of prob- right one increases from h b to h hU at x xU 1. One ability, and (A) represents the probability of the event A. must assume a b if the central part is to be weakly un- Probability considerations permit one to express Eqs. 9 ­ stable. Then, let us decompose the space GP into the equiva- 11 in the following useful form. Let us consider for sim- lence classes ab containing all the generating functions plicity the case where the ensemble is initially in the lower characterized by the same a and b, and let us express Eq. saturation state. This eliminates from all equations the com- 12 as an integral over those equivalence classes, that is, plicated dependence on the initial state g (h0) of the indi- vidual systems. In particular, Eq. 11 can be written as X t x g ht ;xn hn ,...,h1 ;xL ...d g da db. a b ab 14 Xt x g ht ;xn hn ,...,h1;xL ...d g . 12 G As discussed at the end of Sec. II A all generating functions characterized by the same set of weakly unstable intervals At the end of the preceding section, we mentioned the fact and differing only in the values they take inside these inter- that the input reversal sequence h vals give rise to identical hysteresis properties. This means 1 ,h2 ,...,hn selects a se- quence of weakly stable portions of the generating function. that the function x g appearing in Eq. 14 takes the same Let us denote by p(x values for any g t ,ht ;hn ,...,h1)dxt the probability of ab , so it can be taken out of the integral. having a function g G such that g(u) h We obtain t for some u in the interval xt,xt dxt and such that there exists a sequence of x values, x1 ,x2 ,...,xn , for which g(x1) h1 ,g(x2) X h t ab ht a,b da db, 15 2 ,...,g(xn) hn and xL ,x1 , x1 ,x2 ,..., xn 1 ,xn , a b xn ,xt are all weakly stable subintervals. Then one can for- mally write Eq. 12 in the equivalent form where ab ht expresses in simplified operator form the de- pendence of x g on a, b, and input history, whereas (a,b) represents the measure of the class ab . It is easy to check X through Figs. 3 and 5 that t xt p xt ,ht ;hn ,...,h1 dxt . 13 ab ht 1. In other words, ab ht is a rectangular-loop operator with switching inputs PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1433 a and b. The hysteresis model is a weighted superposition of these operators, which means that it is precisely the Preisach model. In the next section, we will show that the Preisach model can also emerge in a completely different context, when the generating functions g(x) are interpreted as sample functions of a Markovian stochastic process. III. MEASURES GENERATED BY STOCHASTIC PROCESSES The main difficulty of the formulation discussed in the preceding section lies in its abstract nature. One needs some tools to generate and manipulate the measure before one can apply the approach to specific situations of interest. In this section, we discuss the case where this issue is addressed FIG. 6. Left: example of stochastic process sample functions by interpreting the generating functions g(x) as sample func- involved in the study of level crossing through the boundary h a tions of some stochastic process. We will show that, quite thick line or h b thin line , starting from the initial condition remarkably, the calculation of Eq. 13 can then be reduced x x0 at h h0 . Right: same as before, in the particular case where to the solution of the level-crossing problem also called exit h0 b and crossing through h a only is considered. problem or first-passage-time problem 26,27 for the sto- passage-time problem for the stochastic process 26,27 . Let chastic process considered. This will create a direct bridge us restrict the level-crossing analysis to the sample functions between two such distant fields as the theory of hysteresis that reach the upper boundary h a first, and let us take the and the theory of stochastic processes, and will permit us to limit h0 b as shown in Fig. 6 b . If we interpret the func- exploit the machinery of level-crossing analysis to derive tion shown in Fig. 6 b as a portion of some generating func- analytical results on hysteresis. In particular, we will show tion extending outside the interval xb ,xa , we immediately that homogeneous continuous Markovian processes give rise recognize that the interval xb ,xa is weakly stable see Sec. to Preisach-type hysteresis and we will derive explicit ana- II A, paragraph f because g(xb) b, g(xa) a, and b lytical expressions for the associated Preisach distribution g(x) a for any x in the range xb x xa . Therefore, xa (a,b). and xb are admissible reversal outputs that may be encoun- tered under input histories with input reversals at h a and h b, and the solution of the particular level-crossing prob- A. Markovian processes with continuous sample functions lem shown in Fig. 6 b is accordingly expected to give direct Let us consider the stochastic process g information about the probability distribution of those rever- x . To avoid con- fusion, we point out that the independent variable x has noth- sal outputs. ing to do with the real time t: it will play the role of a To analyze in detail the consequences of this idea, let us fictitious time to be eventually identified with the system assume that the particular level-crossing problem of Fig. 6 b output. We assume that the process is Markovian, that is, its has been solved, so that we know the conditional probability evolution under given initial conditions, say at x x density T(a,x 0 , de- a b,xb) of having a level-crossing event at x pends on these conditions only and not on the behavior of the xa that is, of having g(xa) a] conditioned by the fact process for x x that g(x 0 . In addition, we assume that the process is b) b. The function T is defined for a b and is a diffusion one, which means that almost all its sample nonanticipating, that is, T(a,xa b,xb) 0 for xa xb . It functions g(x) are continuous functions of x. We will use the obeys the normalization condition letter h, with appropriate subscripts, to denote values taken by these sample functions. T a,xa b,xb dxa 1. 16 In Sec. II A, paragraph f , we discussed the fact that, xb given the generating function g(x), any arbitrary input re- versal sequence h The quantities xa and xb are in general random variables. Let 1 ,h2 ,...,hn selects a sequence of weakly stable portions of that function. We will show now that, us denote by pa(xa) and pb(xb) their probability distribu- when g(x) is interpreted as a sample function of g tions. These distributions are not independent, because they x , weakly stable intervals are naturally and intimately related to the must satisfy the equation solution of the level-crossing problem for gx . To this pur- xa pose, let us consider the interval b,a of the h axis and let pa xa T a,xa b,xb pb xb dxb . 17 us select in it the point h 0 , with b h0 a. Let us imagine that we generate a sample function g(x) of the process start- Notice that, because of the Markovian character of the pro- ing from (h0 ,x0), and that we follow it until it reaches one cess, Eq. 17 is fully independent of the behavior of the of the two boundaries, h b or h a, for the first time Fig. process outside the interval xb ,xa . Let us define the space 6 a . The value of x at which g(x) reaches the boundary is G of Sec. II B as the space containing all those sample func- a random variable. The problem of determining the statistical tions of the given Markovian process which satisfy the re- properties of this random variable is known in the literature quirements of Eq. 2 for some xL and xU , that is, g(xL) as the level-crossing problem or exit problem or first- hL , g(xU) hU , and hL g(x) hU for any x in the range 1434 G. BERTOTTI, I. D. MAYERGOYZ, V. BASSO, AND A. MAGNI PRE 60 Equation 18 can be also used to calculate the probability distribution p1(x1 ;h1) of the reversal output x1 at h1 . We find x1 p1 x1 ;h1 T h1,x1 hL ,xL pL xL dxL , 20 hL h1 hU . Similar considerations apply to the second, decreasing input branch, where ht decreases from h1 to h2 and xt accordingly decreases from x1 to x2 see Fig. 7 b . The weakly stable interval to consider is now xt ,x1 . By applying Eq. 17 to this interval, one obtains x1 p1 x1 ;h1 T h1,x1 ht ,xt p xt ,ht ;h1 dxt , 21 h2 ht h1 . The main difference with respect to Eq. 18 is that the un- FIG. 7. Level-crossing problems to be solved to calculate hys- known distribution p(xt ,ht ;h1) is now inside the integral on teresis in the Markovian process. Top field increasing from h the right-hand side of Eq. 21 , so Eq. 21 is actually an L toward h integral equation for p(x 1): the level crossing must be considered in the interval t ,ht ;h1). It is this difference in the hL ,ht , with known distribution pL(xL) at the lower h boundary structure of Eqs. 18 and 21 that is responsible for the hL . Bottom field decreases from h1 toward h2 , not shown : the onset of hysteresis in the average output. The comparison of level crossing must be considered in the interval ht ,h1 , with Fig. 7 a with Fig. 7 b gives a pictorial illustration of this known distribution p1(x1 ;h1) at the upper h boundary h1 . difference. Although the probability distributions of xL and x1 are the same, the level-crossing problems to solve under xL x xU . In general, xL and xU will be random variables, increasing or decreasing input are different, and therefore taking different values for each g G. Then, let us study the give rise to different probability distributions and different evolution of the ensemble described by Eq. 12 , assuming average outputs. that the ensemble is initially in the lower saturation state The procedure that we have described can be continued to i.e., h(0) hL]. Let us denote by h1 ,h2 ,...,hn the alternat- calculate the distribution p2(x2 ;h2 ,h1) of the second rever- ing sequence of dominant input extrema controlling the evo- sal output, given by the solution of the integral equation lution of the ensemble this sequence was indicated as h x M1 ,hm1 ,hM2 ,hm2 ,... in Sec. II A, paragraph e . Let us 1 analyze in some detail what happens when the input h p1 x1 ;h1 T h1,x1 h2,x2 p2 x2;h2,h1 dx2, t in- creases from hL up to h1 along the first hysteresis branch, 22 and then decreases from h1 to h2 along the second one. We hL h2 h1 denote by xt the output value associated with ht for a given generating function see Fig. 7 a . The interval xL ,xt is and then the distributions p(xt ,ht ;h2 ,h1), weakly stable for each g G, so we can apply Eq. 17 , with p3(x3 ;h3 ,h2 ,h1), and so on up to the distribution b hL , xb xL , a ht , xa x p g (ht ;xL) xt , pb(xb) n(xn ;hn ,...,h1) of the last reversal output. At this point, pL(xL), pa(xa) p(xt ,ht): the probability density p(xt ,ht ;hn ,...,h1) of the current out- put at time t is given-depending on whether the current xt input is increasing or decreasing-by one of the following p xt ,ht T ht ,xt hL ,xL pL xL dxL , hL ht h1. two equations: 18 p xt ,ht ;hn ,...,h1 The probability distribution pL(xL) of the lower saturation xt output x T h L can be chosen at will; it is part of the characteriza- t ,xt hn ,xn pn xn ;hn ,...,h1 dxn , tion of the initial state of the ensemble. After that, Eq. 18 permits one to calculate the unknown distribution p(xt ,ht) h on the basis of the known functions p n ht hn 1 , L(xL) and T(h 23 t ,xt hL ,xL). The distribution p(xt ,ht) is exactly the p function needed in Eq. 13 to calculate the average response n xn ;hn ,...,h1 of the system, according to the expression xn T hn ,xn ht ,xt p xt ,ht ;hn ,...,h1 dxt , Xt ht xtp xt ,ht dxt . 19 hn 1 ht hn , PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1435 and the corresponding average output is branches Eq. 29 . The importance of this result lies in the fact that it implies the validity of the so-called congruency X property 9 . It is known that return-point memory built in t ht ;hn ,...,h1 xtp xt ,ht ;hn ,...,h1 dxt . 24 the description from the beginning, see Sec. II A, paragraph e and congruency represent the necessary and sufficient By the analysis just concluded, we have reduced the original conditions for the description of a given hysteretic system by functional integral over G Eq. 12 to a chain of integrals the Preisach model 28,9,12 . Therefore, we conclude that and integral equations Eqs. 18 , 20 , 22 , and 23 , de- the hysteresis generated by a homogeneous, diffusion Mar- pendent on the saturation distribution pL(xL) arbitrarily kovian process is of Preisach type. The process is fully de- chosen and the transition density T(a,xa b,xb). The central scribed by the function XT(a b) Eq. 27 , which is nothing problem is then the calculation of T(a,xa b,xb) for a given but the Everett function associated with the Preisach descrip- process. tion. The function XT(a b) represents the average value of x at which the generating function crosses the level h a for B. Homogeneous processes the first time, starting at x 0 from the initial level h b see Particularly simple and interesting results are obtained Fig. 6 b . The description of hysteresis is reduced to the when the statistical properties of the process considered are solution of this particular level-crossing problem for the sto- translationally invariant with respect to x, that is, when the chastic process. process is homogeneous with respect to x. In fact, in this case Remarkably, this solution can be worked out in closed it is not necessary to determine the complete function analytical form. To this end, let us start from the description T(a,x of the process in terms of its Ito stochastic differential equa- a b,xb) in order to predict the hysteresis properties of the ensemble. To clarify this point, let us come back to the tion 26 , first of Eqs. 23 . Because of the assumed homogeneity of dh A h dx B h dW the process, T(a,x x , 30 a b,xb) T(a,xa xb b,0). Therefore, where dW p x represents the infinitesimal increment of the x,h;hn ,...,h1 Wiener process W(x), A and B are independent of x because x of the assumed homogeneity of the process, and x plays the T h,x xn hn,0 pn xn ;hn ,...,h1 dxn , role of time. The statistics of the process are fully described by the transition density P(h,x h0 ,x0), giving the probabil- 25 ity density that a sample function of the process takes the value h at the position x, conditioned to the fact that it takes where we have dropped for simplicity the t subscript in x and the value h0 at x x0 . The Fokker-Planck equation for the h. According to Eq. 24 , the average system response is transition density associated with Eq. 30 , P(h,x h0) obtained by multiplying both members of Eq. 25 by x and P(h,x h0,0), is by integrating over x. By expressing x as x (x xn) xn and by rearranging the appropriate integrals on the right- hand side, we obtain x P h,x h0 h A h P h,x h0 X h;hn ,...,h1 Xn XT h hn , h hn , 26 1 2 2 h2 B2 h P h,x h0 0. 31 where As discussed before, the situation of interest is the one de- X picted in Fig. 6. The process starts, at x x T a b uT a,u b,0 du, a b 27 0 0, from h 0 h0 . We wish to determine the statistics of the x value at which the process reaches the level h a for the first time, in and the limit h0 b. This is obtained by solving Eq. 31 under the initial condition P(h,0 h0) (h h0), together with the Xn xnpn xn ;hn ,...,h1 dxn . 28 assumption of absorbing boundary conditions at h a and h b, and then by taking the limit h0 b. The mathematical details of the analysis are discussed in Appendix C. The When the second of Eqs. 23 is the relevant equation, by perfectly similar considerations one obtains solution for XT(a b), expressed in terms of the function u A u X h;hn ,...,h1 Xn XT hn h , h hn . 29 u exp 2 du , 32 0 B2 u We see that the hysteresis properties of the system are fully controlled by the first moment of T only, given by Eq. 27 . reads Equation 26 shows that the shape of a generic ascending 2 a u hysteresis branch starting from the reversal field hn is the X u du same regardless of the past input history. The influence of T a b K b,a b b past history is summarized in the value of Xn Eq. 28 , and the branches generated by different histories differ by a mere a u du du , 33 shift along the X axis. The same is true for descending u B2 u u 1436 G. BERTOTTI, I. D. MAYERGOYZ, V. BASSO, AND A. MAGNI PRE 60 where a K b,a u du. 34 b C. Preisach distribution associated with a given homogeneous process The quantity XT(a b) given by Eq. 33 coincides with the Everett function of the Preisach model associated with the homogeneous stochastic process. Therefore, as discussed in 9 , the Preisach distribution (a,b) is given by 1 2X a,b T a b 2 a b . 35 By deriving Eq. 33 , one finds a b FIG. 8. Typical hysteresis curves for the Wiener process, calcu- a u a,b 2 u du lated from Eq. 39 . K3 b,a b b The Preisach distribution depends on the difference (a b) a only, and tends to the value 1 u du du , 36 3 when ( a b ) 0, in agreement u B2 u u with Eq. 39 . Typical hysteresis branches calculated from Eq. 41 are shown in Fig. 9. that is, taking into account Eq. 33 , (c) Ornstein-Uhlenbeck process. In this case, A(h) h/ , B(h) 1, with 0. We find a b a,b X K2 T a b . 37 u exp u2/ , b,a 42 Let us calculate the Preisach distribution associated with 3 a2 3 b2 K b 1 , some typical stochastic processes. b,a a 12 , 2 ; 2 , 2 ; (a) Wiener process. The Wiener process is described by A(h) 0, B(h) 1 see Eq. 30 . Therefore, we obtain from where (a,c;x) is the confluent hypergeometric function. Eqs. 32 and 34 , The Preisach distribution and XT(a b) are obtained by in- serting these expressions into Eqs. 33 , 36 , and 37 . u 1, 38 K IV. CONCLUSIONS b,a a b. The formulation developed in the previous sections is By inserting these expressions into Eqs. 33 , 36 , and 37 , general enough to offer various possibilities for further stud- we find ies and applications. From the mathematical viewpoint, the a,b 13 , 39 XT a b 13 a b 2. The Preisach distribution is simply a constant and all hyster- esis branches are parabolic Fig. 8 . (b) Wiener process with drift. By this, we mean the case where by A(h) 1/(2 ), B(h) 1, with 0. We have u exp u/ , 40 K b,a exp b/ exp a/ . The Preisach distribution and XT(a b) are given by x coth x 1 a,b sinh2 x , 41 a b X FIG. 9. Typical hysteresis curves for the Wiener process with T a b 4 x coth x 1 , x 2 . drift, calculated from Eq. 41 . PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1437 basic issue is the role of return-point memory in the func- U g h,x0 x x0 ,xU :g x h . A1 tional integration approach developed in Sec. II. We know that return-point memory is inherent in the formulation, and Because of the continuity of g, U g is closed and will there- it is natural to ask under what additional conditions, if any, fore contain its minimum xm . This is the smallest x an arbitrary scalar hysteretic system exhibiting return-point x0 ,xU for which g(x) h. Because g(x0) h, then memory can be described through Eqs. 10 ­ 13 , by choos- g(x) h for any x x0 ,xm), i.e., max g(u):u x0,xm ing appropriately the space G and the measure . For the g(xm) h. Therefore, the equilibrium state xm is such that moment, we do not have a general answer to this basic ques- h g (xm ;x0) h. The proof of the case hL h g(x0) is tion. analogous. The only difference is that one must consider the From the physical viewpoint, the results obtained in the maximum xM of the set case where the measure is associated with a stochastic pro- cess are of direct interest to all those situations where some L g h,x0 x xL ,x0 :g x h . A2 dominant degree of freedom, say X, evolves in a random free-energy landscape, and the associated dynamics are over- Following the results of theorem 1, let us introduce the func- damped. By this we mean that X obeys an equation of the tion x g (h;x0) defined as follows: form if h x L h g x0 g h;x0 max L g h,x0 . dX F X min U g h,x0 if g x0 h hU A3 dt H t X , 43 The function x g (h;x0) is the inverse of h g (x;x0). In fact, where H(t) is the time-dependent driving field, F(X) is the according to theorem 1, h g x g (h;x0);x0... h see Fig. 2 . free energy of the system, and 0 is some typical friction It monotonically increases with h and, as a rule, it is not constant. Under small enough field rates, the solutions of Eq. continuous in h. However, it is continuous in x0 , because 43 -once expressed in terms of H as a function of both max L g and min U g are continuous in x0 , as a conse- X-precisely approach the behavior shown in Fig. 1 12,29 , quence of the continuity of g. Notice that the graph of x g so that our formulation can be directly applied, if one knows consists uniquely of equilibrium states, that is, of points of the statistical properties of the free-energy gradient F/ X. the generating function g(x). A particularly important example is the motion of magnetic domain walls in ferromagnets, where Eq. 43 often provides APPENDIX B: ORDERING PROPERTIES OF T g h... a good physical description, and various forms of structural disorder point defects, dislocations, grain boundaries, etc. As discussed in Sec. II A, given the equilibrium states are responsible for the random character of F/ X. There s1 (h1 ,x1) and s2 (h2 ,x2) we say that s1 s2 if x1 x2 in are a series of classical papers in the literature 30,31 , where the usual sense. The set of equilibrium states is totally or- the domain wall picture has been applied to the prediction of dered with respect to this relation. Before considering the coercivity and magnetization curve shapes, starting from theorems deriving from the existence of this ordering rela- some assumption about the properties of F(X). Equations tion, let us prove the following four lemmas, involving the 33 and 36 provide a general solution for the case where sets U g and L g defined by Eqs. A1 and A2 . the process F/ X is Markovian, continuous, and homoge- Lemma 1. Given xA xB and hL h min g(xA),g(xB) , neous. In particular, the proven equivalence of Markovian then max L g (h,xA) max L g (h,xB). In fact, L g (h,xB) disorder to the Preisach model gives a sound statistical inter- L g (h,xA) LAB , where LAB x (xA ,xB] :g(x) h . pretation of the latter in terms of stochastic dynamics in LAB is empty or contains elements that are all greater than quenched-in disorder. In this respect, the extension of the any element of L g (h,xA). In both cases, the lemma is analysis of Sec. III to non-Markovian and/or nonhomoge- proven. neous processes would be of definite interest, as a way to Lemma 2. Given xA xB and hU h max g(xA),g(xB) , provide quantitative predictions of hysteresis features under then min U g (h,xA) min U g (h,xB). In fact, U g (h,xA) more realistic conditions and to indicate in what direction U g (h,xB) UAB , where UAB x xA ,xB):g(x) h . one should generalize the Preisach model in order to improve UAB is empty or contains elements that are all smaller than the macroscopic description of hysteresis generated by vari- any element of U g (h,xB). In both cases, the lemma is ous forms of structural disorder. proven. Lemma 3. Given xA xB and g(xA) h g(xB), then APPENDIX A: PROPERTIES OF h max L g (h,xA) min U g (h,xB). In fact, given any x g x;x0... AND x g h;x0... L g (h,xA) and x U g (h,xB), x xA xB x . This will The function h g (x;x0) defined by Eq. 3 is continuous hold in particular for x max L g (h,xA) and x and nondecreasing with respect to x and it is continuous with min U g (h,xB), which proves the lemma. respect to x0 . Its main properties derive from the following Lemma 4. Given xA xB and g(xA) h g(xB), then theorem. min U g (h,xA) max L g (h,xB). In fact, the set ZAB x Theorem 1. Given any h hL ,hU , there exists at least xA ,xB :g(x) h is not empty, which means that one equilibrium state x xL ,xU , such that h g (x;x0) h. min U g (h,xA) min ZAB and max L g (h,xB) max ZAB . Be- Proof. The theorem holds by definition for h g(x0), h cause min ZAB max ZAB , the lemma is proven. hL , and h hU . Then, let us consider the case g(x0) h Theorem 2. Given the equilibrium states sA and sB , sA hU . Let us introduce the set sB , then T g (h)sA T g (h)sB for any h hL ,hU . 1438 G. BERTOTTI, I. D. MAYERGOYZ, V. BASSO, AND A. MAGNI PRE 60 Proof. Let us express the states as sA hA g(xA),xA... The total probability b,a (h0) that the process leave the and sB hB g(xB),xB..., with xA xB . According to Eq. interval b,a through h a is given by the expression 4 , the application of T g (h) changes xA and xB into x g (h;xA) and x g (h;xB). By applying lemmas 1­4 to the J a,u h definition of x b,a h0 0 du. C4 g (h;x0) Eq. A3 , one finds that 0 x g (h;xA) x g (h;xB) for any h hL ,hU . Theorem 3. Given the equilibrium states s By integrating Eq. C3 from x 0 to , and by taking into A and sB , sA s account that J(a,0 h B , and the inputs hA and hB , hA hB , then T g (hA)sA 0) J(a, h0) 0, one finds that T g (hB)sB . b,a (h0) satisfies the differential equation Proof. Let us express the states as sA hA g(xA),xA... and s d2 b,a d b,a B hB g(xB),xB..., with xA xB . From theorem 2, we 1 0 C5 have that x 2 B 2 h 0 dh2 A h0 dh g (hA ;xA) x g (hA ;xB). On the other hand, 0 0 x g (hA ;xB) x g (hB ;xB), because of the monotonicity of x with the boundary conditions g (h;x0) with respect to h. Therefore, x g (hA ;xA) x g (hB ;xB). Theorem 4. Let us consider the initial equilibrium states b,a a 1, sA(0) and sB(0), sA(0) sB(0), and let us apply to them the two ordered input histories h b,a b 0, C6 A(t) hB(t). Then, at any sub- sequent time t 0, sA(t) sB(t). deriving from the fact that the process will certainly cross the Proof. Let us consider the evolution of the two states boundary h a if it starts from that same level, that is, from sA(t) and sB(t) first from t 0 up to the time of the first h reversal of h 0 a, whereas it will never reach h a if it starts from A(t) or hB(t), then from this time to the time of h the second reversal of h 0 b, because in that case it will certainly cross the bound- A(t) or hB(t), and so on. By consid- ary h b first. The solution of Eq. C5 is then ering that initially sA(0) sB(0) and by applying theorem 3, we find that order is preserved in the first interval and that 1 h0 the states at the end of the interval are still ordered. This b,a h0 K u du, C7 permits one to conclude that order is preserved also in the b,a b second interval, and so on. where (u) and K b,a are given by Eqs. 32 and 34 , re- spectively. APPENDIX C: SOLUTION OF LEVEL-CROSSING The probability density p b,a (x h0) that the process PROBLEM reaches the boundary h a at the position x is given by The probability current associated with Eq. 31 is J a,x h p 0 b,a x h0 C8 1 b,a h0 J h,x h0 A h P h,x h0 2 h B2 h P h,x h0 . and the mean value x C1 b,a (h0) of the level-crossing position is The rate at which the process leaves the interval b,a start- ing from h h0 at x 0 is obtained by integrating Eq. 31 x b,a h0 up b,a u h0 du over h. One obtains 0 1 a uJ a,u h0 du. C9 x P h,x h0 dh J a,x h0 J b,x h0 . b,a h0 0 b C2 By definition, the conditional probability density T(a,x b,0) Equation C2 shows that the probability current at the of Eq. 27 is given by the limit of Eq. C8 for h0 b, that boundaries is just proportional to the probability density that is, a level-crossing event takes place at the position x. As men- T a,x b,0 p tioned before, we are interested in level crossing through the b,a x b , C10 upper boundary h a, described by the probability current and the function XT(a b) defined by Eq. 27 is accordingly J(a,x h0). According to Eq. C1 , the functional dependence given by of J(a,x h0) on x and h0 is the same as that of P(a,x h0) P(a,x h X 0,0) P(a,0 h0 , x). This means that J(a,x h0) T a b x b,a b . C11 obeys the backward equation 24 By multiplying Eq. C3 by x, by integrating it from x 0 to , by taking into account that J(a,0 h0) J(a, h0) 0, x J a,x h0 A h0 h J a,x h0 and by making use of Eq. C4 , one finds that the function 0 2 f b,a h0 b,a h0 x b,a h0 C12 12 B2 h0 h2 J a,x h0 0. C3 0 obeys the differential equation PRE 60 FUNCTIONAL INTEGRATION APPROACH TO HYSTERESIS 1439 d2f d f b b 1 b,a b,a df b,a x X 2 B 2 h 0 A h b,a b T a b . dh2 0 b,a h0 0 dh K K 0 dh0 0 h b,a b,a 0 b C13 C17 with the boundary conditions Equations C16 and C17 permit one to write Eq. C15 in the form f b,a a f b,a b 0 C14 deriving from the fact that x b,a (a) 0 by definition, d f b,a h0 b,a u whereas h 2 du . b,a (b) 0 because of Eq. C6 . Equation C13 is dh 0 XT a b K B2 u u a linear, nonhomogeneous first-order differential equation for 0 b,a b C18 d f b,a /dh0 . The solution reads d f b,a h0 b,a u By integrating Eq. C18 from b to h h du , C15 0 and by inverting the dh 0 C 2 order of integration in the double integral, one obtains 0 b B2 u u where b,a (u) and (u) are given by Eq. C7 and Eq. 32 , respectively. The constant of integration C can be ex- f b,a h0 XT a b b,a h0 pressed as 2 h0 u K u du b,a b b 1 C df b,a h0 . C16 b dh0 u du du , C19 h0 b a B2 u u By deriving Eq. 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