Phase transitions. Phase transitions Measuring instantaneous values ​​using the delta function

Dirac delta function

The delta function (5-function) was introduced by the English physicist P. A. M. Dirac “out of necessity” when he created the mathematical apparatus quantum mechanics. Mathematicians “did not recognize” it for some time, after which they created the theory of generalized functions, a special case of which is the δ-function.

According to the (naive) definition, the δ-function is zero everywhere except one point, but the area covered by this function is equal to one:

These contradictory

requirements cannot be satisfied by a "normal" type function.

Zeldovich Ya.B. Higher mathematics for beginning physicists and technicians. -M.: Nauka, 1982.

Actually like a differential δх is not a number (equal to zero), and the phrase “infinitesimal quantity” is difficult to understand qualitatively, to understand correctly δх not as a number, but as a limit (process), and the δ-function can also be correctly understood as a limit (process). In Fig. 3.7.1 and 3.7.2 show several functions (depending on the parameter), the limit of which is the δ-function. There are an infinite number of such functions - everyone can choose their own.

The δ-function has many useful properties, being, in particular, a continuum analogue of the Kronecker symbol δкк

compare with

Another amazing relationship shows how to differentiate by integrating:

Where 8 - derivative 8- functions.

Rice. 3.7.1 - Two successive approximations to δ-

Dirac functions. Feature shown

Rice. 3.7.2 - Two functions that are in the limit A ->∞ give δ-functions:

Finally, note that the interval from the δ-function:

Where in(x)- Heaviside function,

step, with a break at a point x = 0 .

Phase transitions

In order to talk about phase transitions, it is necessary to define what phases are. The concept of phases occurs in a variety of phenomena, so instead of giving general definition(the more general it is, the more abstract and obscure it is, as it should be), let’s give a few examples.

First, an example of their physics. For the ordinary, most common liquid in our lives - water, three phases are known: liquid, solid (ice) and gaseous (steam). Each of them is characterized by its own parameter values. The important thing is that when external conditions change, one phase (ice) transforms into another (liquid). Another favorite object of theorists is ferromagnets (iron, nickel and many other pure metals and alloys). At low temperatures (for nickel below T= 3600 WITH) a nickel sample is ferromagnetic, when removing the outer magnetic field it remains magnetized, i.e. can be used as a permanent magnet. At temperatures above Ts this property is lost; when the external magnetic field is turned off, it goes into a paramagnetic state and is not a permanent magnet. When the temperature changes, a transition occurs - phase transition- from one phase to another.

Let us give another geometric example from the theory of percolation. Randomly cutting out bonds from the network, eventually when the concentration of the remaining bonds is R will be less than a certain value rs, it will no longer be possible to walk along the grid “from one end to the other.” Thus, the mesh from the state of flow - the "leaking" phase - will go into the state of the "non-leaking" phase.

From these examples it is clear that for each of the systems considered there is a so-called order parameter that determines which phase the system is in. In ferromagnetism, the order parameter is magnetization in a zero external field; in percolation theory, it is the connectivity of the network, or, for example, its conductivity or the density of an infinite cluster.

There are different types of phase transitions. First-order phase transitions are those in which several phases can simultaneously exist in a system. For example, at a temperature of 0° C ice floats in water. If the system is in thermodynamic equilibrium (there is no heat supply or removal), then the ice does not melt or grow. For second-order phase transitions, the simultaneous existence of several phases is impossible. A piece of nickel is either in a paramagnetic state or in a ferromagnetic state. A mesh with randomly cut connections is either connected or not.

Decisive in the creation of the theory of phase transitions of the second order, which was initiated by L.D. Landau, there was an introduction of the order parameter (we will denote it G]) as a distinctive feature of the system phase. In one of the phases, for example, paramagnetic, r] = 0, and in the other, ferromagnetic, G ^ 0. For magnetic phenomena, the order parameter ] is the magnetization of the system.

To describe phase transitions, a certain function of parameters that determine the state of the system is introduced - G(n, T,...). IN physical systems This is Gibbs energy. In each phenomenon (percolation, a network of “small worlds”, etc.) this function is determined “independently”. The main property of this function, the first assumption of L.D. Landau - in a state of equilibrium, this function takes a minimum value:

In physical systems we talk about thermodynamic equilibrium, in the theory of complex chains we can talk about stability. Note that the minimality condition is determined by varying the order parameter.

The second assumption of L.D. Landau - during a phase transformation n = 0. According to this assumption, the function b(n,T,...) near the phase transition point can be expanded into a series in powers of the order parameter n:

where n = 0 in one phase (paramagnetic, if we are talking about magnetism and incoherent, if we are talking about a grid) and n ^ 0 in another (ferromagnetic or connected).

From the condition

which gives us two solutions

For T > Tc the solution n = 0 must take place, and for T< Тс solution n ^ 0. This can be satisfied if for the case T > Tc and n = 0 select A > 0 . In this case, there is no second root. And for the occasion T < Ts there must be a second solution, i.e. must be fulfilled A< 0. Thus:

A > 0 at T > Tc, A< 0 at T< Тс ,

Landau's second assumption requires the fulfillment of A(Tc) = 0. The simplest form of function A(T) that satisfies these requirements is

The so-called critical index, and the function C(g],T) takes the form:

In Fig. 3.8.1 shows the dependence b(n, T) for T > Tc And T< Тс .

Rice. 3.8.1 - Parameter Function Graphs G(n, T) For T > Tc And T< Тс

Poston T., Stewart I. Catastrophe theory and its applications. - M.: Mir, 1980. Gilmore R. Applied theory of disasters. - M.: Mir, 1984.

Qualitative dependence of parameters G(j], T) on the order parameter ] is shown in Fig. 3.8.1 (G0 = 0). The dependence of the order parameter ] on temperature is shown in Fig. 3.8.2.

A more advanced theory takes into account that when T > Tc the order parameter ], although very small, is not exactly equal to zero.

Transition of the system from state with h = 0 at T > Tc in a state with h- 0 when decreasing T and reaching values T £ Tc can be understood as a loss of position stability h = 0 at T £ Tc. Recently a mathematical theory appeared

with the sonorous name “The Theory of Disasters”, which describes many different phenomena from a single point of view. From the point of view of catastrophe theory, a second-order phase transition is an “assembly catastrophe.”

Rice. 3.8.2 - Order parameter dependence n from temperature: at T< Tc and nearby Tc order parameter n Behave like power function, and when T> Tc n = 0

DELTA FUNCTION

Definition. Delta function

(2.1)

A generalized function

Fig.1. Delta function

Normalization condition

, . (2.2)

a, as shown in Figure 1, b

Function parity follows from (2.1)

. (2.2a)

, (2.2b)

as follows from Fig. 1, b.

Orthonormality. Lots of features

Properties of the DELTA FUNCTION

Filtering property

we get

b, we find

,

, . (2.5)

Orthonormality of the basis

In (2.5) we assume

, ,

. (2.7)

Performed

,

, (2.8)

Proof

Simplifying the Argument

If are the roots of the function , Then

. (2.9)

Proof

.

In a small neighborhood we expand Taylor series

and limit ourselves to the first two terms

Let's use (2.8)

Let's compare integrands and we obtain (2.9).

Convolution

From the definition of convolution (1.22)

,

at we get

.

We believe , and we find

.. (2.35a)

and (2.35a) give

. (2.35b)

we get

. (2.36a)

and (2.36a) give

. (2.36b)

. (2.37a)

we get

. (2.37b)

Comb function

(2.53)

Models unlimited crystal lattice, antenna and other periodic structures.

With the Fourier transform, a comb function becomes a comb function.

,

(2.8)

we get

. (2.54)

Properties

Even function

,

periodic

,

period . The filtering property of delta functions gives

. (2.55)

Fourier image

For periodic function with period L The Fourier image is expressed in terms of Fourier coefficients

, (1.47)

, (1.49)

For a comb function with period we obtain

,

where the filtering property of the delta function is taken into account. From (1.47) we find the Fourier transform

. (2.56)

The Fourier transform of a comb function is the comb function.

From (2.56), using the Fourier theorem on the scaling transformation of the argument, we obtain

. (2.59)

Increasing the period of the comb function ()reduces the period and increases the amplitude of its spectrum .

Fourier series

We use

For , we get

DELTA FUNCTION

Definition. Delta function

models a point disturbance and is defined as

(2.1)

The function is equal to zero at all points except where its argument is equal to zero, and where the function is infinite, as shown in Fig. 1, A. Setting the values ​​at the points of the argument is ambiguous due to its turning to infinity, so the delta function is generalized function , and requires further definition in the form of normalization.

Fig.1. Delta function

Normalization condition

, . (2.2)

The area under the graph of a function is equal to one in any interval containing a point a, as shown in Figure 1, b. Therefore, the delta function models a point disturbance of unit magnitude.

Function parity follows from (2.1)

. (2.2a)

From symmetry about a point we obtain

, (2.2b)

as follows from Fig. 1, b.

Orthonormality. Lots of features

forms an orthonormal infinite-dimensional basis.

The delta function was used in optics by Kirchhoff in 1882, and in electromagnetic theory by Heaviside in the 90s of the 19th century.

Gustav Kirchhoff (1824–1887) Oliver Heaviside (1850–1925)

Oliver Heaviside, a self-taught scientist, was the first to use vectors in physics, developed vector analysis, introduced the concept of an operator and developed operational calculus– operator solution method differential equations. He introduced the switching function, which was later named after him, and used a point impulse function - the delta function. Applied complex numbers in the theory of electrical circuits. For the first time he wrote down Maxwell's equations in the form of 4 equalities instead of 20 equations, as Maxwell had. Introduced terms: conductivity, impedance, inductance, electret . He developed the theory of long-distance telegraph communication and predicted the presence of an ionosphere near the Earth - the Kennelly-Heaviside layer.

The mathematical theory of generalized functions was developed by Sergei Lvovich Sobolev in 1936. He was one of the founders of the Novosibirsk Academic Town. The Institute of Mathematics of the SB RAS is named after him.

Sergei Lvovich Sobolev (1908–1989)

Properties of the DELTA FUNCTION

Filtering property

For a smooth function without discontinuities, from (2.1)

we get

Assuming , and using the delta function in the form of a limit at , shown in Fig. 1, b, we find

,

Integration gives the filtering property in integral form

, . (2.5)

Orthonormality of the basis

In (2.5) we assume

, ,

and we obtain the condition for the orthonormality of a basis with a continuous spectrum

. (2.7)

Argument scaling

Performed

,

, (2.8)

Proof

Let's integrate the product of the delta function with a smooth function over the interval, where:

where a variable replacement is made and the filtering property is used. Comparing the initial and final expressions gives (2.8).

Simplifying the Argument

If are the roots of the function , Then

. (2.9)

Proof

The function is nonzero only near the points, at these points it is infinite.

To find the weight with which infinity enters, we integrate the product with a smooth function over the interval. The contributions are nonzero only in the vicinity of the points

. , (2.10) .. (2.35a)

Fourier's argument shift theorem

and (2.35a) give

. (2.35b)

From (1.1) and integral representation (2.24)

we get

. (2.36a)

Fourier's theorem on the phase shift of a function

and (2.36a) give

. (2.36b)

From (2.35a) and the Fourier differentiation theorem

. (2.37a)

From (2.36a) and the Fourier theorem on multiplication by the argument

we get

. (2.37b)

Introduction

The development of science requires for its theoretical justification more and more “ high mathematics", one of the achievements of which is generalized functions, in particular the Dirac function. Currently, the theory of generalized functions is relevant in physics and mathematics, as it has a number of remarkable properties that expand the capabilities of classical mathematical analysis, expands the range of problems under consideration, and also leads to significant simplifications in calculations, automating elementary operations.

Objectives of this work:

1) study the concept of the Dirac function;

2) consider physical and mathematical approaches to its definition;

3) show the application to finding derivatives of discontinuous functions.

Objectives of the work: to show the possibilities of using the delta function in mathematics and physics.

The paper presents various ways of defining and introducing the Dirac delta function, and its application in solving problems.

Definition of the Dirac function

Basic concepts.

In different questions of mathematical analysis, the term “function” has to be understood with varying degrees of generality. Sometimes continuous but not differentiable functions are considered, in other questions we have to assume that we are talking about functions that are differentiable one or more times, etc. However, in a number of cases the classical concept of function, even interpreted in the broadest sense, i.e. as an arbitrary rule assigning to each value x from the domain of definition of this function a certain number y=f(x) turns out to be insufficient.

Here is an important example: when applying the apparatus of mathematical analysis to certain problems, we have to face a situation where certain analysis operations turn out to be impossible; for example, a function that does not have a derivative (at some points or even everywhere) cannot be differentiated if the derivative is understood as elementary function. Difficulties of this type could be avoided by limiting ourselves to considering only analytical functions. However, such a narrowing of the range of permissible functions is in many cases very undesirable. The need to further expand the concept of function has become especially acute.

In 1930, to solve problems of theoretical physics, the greatest English theoretical physicist P. Dirac, one of the founders of quantum mechanics, did not have enough classical mathematics, and he introduced a new object called the “delta function,” which went far beyond the classical definition of the function .

P. Dirac in his book “Principles of Quantum Mechanics” defined the delta function d(x) as follows:

In addition, the condition is set:

You can clearly imagine a graph of a function similar to d(x), as shown in Figure 1. The narrower you make the strip between the left and right branches, the higher this strip must be in order for the area of ​​the strip (i.e., the integral) to remain its given value, equal to 1. As the strip narrows, we get closer to fulfilling the condition d(x) = 0 at x? 0, the function approaches the delta function.

This idea is generally accepted in physics.

It should be emphasized that d(x) is not a function in the usual sense, since this definition implies incompatible conditions from the point of view of the classical definition of a function and an integral:

at And.

In classical analysis there is no function that has the properties prescribed by Dirac. Only a few years later, in the works of S.L. Sobolev and L. Schwartz, the delta function received its mathematical design, but not as an ordinary, but as a generalized function.

Before moving on to considering the Dirac function, we introduce the basic definitions and theorems that we will need:

Definition 1. An image of the function f(t) or L - an image given function f(t) is a function of the complex variable p, defined by the equality:

Definition 2. Function f(t), defined like this:

called by the Heaviside unit function and is denoted by. The graph of this function is shown in Fig. 2

We'll find L- image of the Heaviside function:

Let the function f(t) at t<0 тождественно равна нулю (рис.3). Тогда функция f(t-t 0) будет тождественно равна нулю при t

To find the image d(x) using an auxiliary function, consider the delay theorem:

Theorem 1. If F(p) is an image of the function f(t), then there is an image of the function f(t-t 0 ), that is, if L(f(t))=F(p), then .

Proof.

By definition of an image we have

The first integral is equal to zero, since f(t-t 0 )=0 at t 0 . In the last integral we make a change of variable t-t 0 =z:

Thus, .

For the Heaviside unit function it was found that. Based on the proven theorem, it follows that for the function, L- the image will be, that is

Definition 3. Continuous or piecewise continuous function d(t,l) argument t, depending on the parameter l, called needle-shaped, If:

Definition 4. Numeric function f, defined on some linear space L, called functionality.

Let us define the set of functions on which the functionals will act. As this collection, consider the set K all real functions c(x), each of which has continuous derivatives of all orders and is finite, that is, it vanishes outside a certain limited area (its own for each of the functions c(x)). We will call these functions main, and their entire set TO - main space.

Definition 5. Generalized function is any linear continuous functional defined on the underlying space TO.

Let's decipher the definition of a generalized function:

1) generalized function f there is functionality on the main functions ts, that is, each ts matches a (complex) number (f, c);

2) functionality f linear, that is, for any complex numbers l 1 And l 2 and any basic functions ts 1 And ts 2 ;

3) functionality f continuous, that is, if.

Definition 6.Pulse- a single, short-term surge in electrical current or voltage.

Definition 7.Average density- body weight ratio m to its volume V, that is .

Theorem 2.(Generalized mean value theorem).

If f(t) is a continuous and is an integrable function on , and does not change sign on this interval, then where.

Theorem 3.Let the function f(x) be bounded on and have at most a finite number of discontinuity points. Then the function is antiderivative for the function f(x) on the interval and for any antiderivative Ф(x) the formula is valid.

Definition 8. The set of all continuous linear functionals defined on some linear space E, forms a linear space. It's called space conjugate With E, and is denoted E * .

Definition 9. Linear space E, in which some norm is specified, is called normalized space.

Definition 10. The sequence is called weakly convergent k, if for each the relation is satisfied.

Theorem 4.If (x n ) is a weakly convergent sequence in a normed space, then there exists a constant number C such that .

Federal Agency for Education

State educational institution of higher professional education
Vyatka State Humanitarian University

Faculty of Mathematics

Department of Mathematical Analysis and Mathematics Teaching Methods

Final qualifying work

Dirac function

Completed by a fifth year student

Faculty of Mathematics Prokasheva E.V.

________________________________/signature/

Scientific adviser:

Onchukova L.V.

signature/

Reviewer:

Senior Lecturer at the Department of Mathematical Analysis and MMM Faleleeva S.A.

________________________________/ signature/

Admitted to defense in the state certification commission

"___" __________2005 Head. department M.V. Krutikhin

Introduction........................................................ ........................................................ ........ 3

Chapter 1. Definition of the Dirac function.................................................... ............. 4

1.1. Basic concepts........................................................ ................................ 4

1.2. Problems leading to the definition of the Dirac delta function………...10

1.2.1. Momentum problem…………………………………………….10

1.2.2.Problem about the density of a material point………………………........11

1.3. Mathematical definition of the delta function………………………..16

Chapter 2. Application of the Dirac function…………………………………………19

2.1. Discontinuous functions and their derivatives………………………………….19

2.2. Finding derivatives of discontinuous functions………………………...21

Conclusion…………………………………………………………………………………25

Introduction

The development of science requires more and more “high mathematics” for its theoretical justification, one of the achievements of which is generalized functions, in particular the Dirac function. Currently, the theory of generalized functions is relevant in physics and mathematics, as it has a number of remarkable properties that expand the capabilities of classical mathematical analysis, expands the range of problems under consideration, and also leads to significant simplifications in calculations, automating elementary operations.

Objectives of this work:

1) study the concept of the Dirac function;

2) consider physical and mathematical approaches to its definition;

3) show the application to finding derivatives of discontinuous functions.

Objectives of the work: to show the possibilities of using the delta function in mathematics and physics.

The paper presents various ways of defining and introducing the Dirac delta function, and its application in solving problems.

Chapter 1

Definition of the Dirac function

1.1. Basic concepts.

In different questions of mathematical analysis, the term “function” has to be understood with varying degrees of generality. Sometimes continuous but not differentiable functions are considered, in other questions we have to assume that we are talking about functions that are differentiable one or more times, etc. However, in a number of cases the classical concept of function, even interpreted in the broadest sense, i.e. as an arbitrary rule assigning to each value x from the domain of definition of this function a certain number y=f(x), turns out to be insufficient.

Here is an important example: when applying the apparatus of mathematical analysis to certain problems, we have to face a situation where certain analysis operations turn out to be impossible; for example, a function that does not have a derivative (at some points or even everywhere) cannot be differentiated if the derivative is understood as an elementary function. Difficulties of this type could be avoided by limiting ourselves to the consideration of analytical functions alone. However, such a narrowing of the range of permissible functions is in many cases very undesirable. The need to further expand the concept of function has become especially acute.

In 1930, to solve problems of theoretical physics, the greatest English theoretical physicist P. Dirac, one of the founders of quantum mechanics, did not have enough classical mathematics, and he introduced a new object called the “delta function,” which went far beyond the classical definition of the function .

P. Dirac in his book “Principles of Quantum Mechanics” defined the delta function δ(x) as follows:

.

In addition, the condition is set:

You can clearly imagine a graph of a function similar to δ(x), as shown in Figure 1. The more

make the strip between the left and right branches thin, the higher this strip must be in order for the area of ​​the strip (i.e., the integral) to maintain its given value equal to 1. As the strip narrows, we get closer to fulfilling the condition δ(x) = 0 at x ≠ 0 , the function approaches the delta function.

This idea is generally accepted in physics.

It should be emphasized that δ(x) is not a function in the usual sense, since this definition implies incompatible conditions from the point of view of the classical definition of a function and an integral:

And .

In classical analysis there is no function that has the properties prescribed by Dirac. Only a few years later, in the works of S.L. Sobolev and L. Schwartz, the delta function received its mathematical design, but not as an ordinary, but as a generalized function.

Before moving on to considering the Dirac function, we introduce the basic definitions and theorems that we will need:

Definition 1. An image of a function f(t) or L - an image of a given function f(t) is a function of a complex variable p, defined by the equality:

, Where M And A– some positive constants.

Definition 2. Function f(t) , defined like this:

, called by the Heaviside unit function and is denoted by . The graph of this function is shown in Fig. 2

We'll find L– image of the Heaviside function:

. (1)

Let the function f(t) at t<0 тождественно равна нулю (рис.3). Тогда функция f(t-t 0) будет тождественно равна нулю при t

To find the image δ(x) using an auxiliary function, consider the delay theorem:

Theorem 1.IfF(p) there is an image of the functionf(t), that is, the image of the functionf(t- t 0 ), that is, ifL{ f(t)}= F(p), That

.

Proof.

By definition of an image we have

Definition. Delta function

,

models a point disturbance and is defined as

(2.1)

The function is equal to zero at all points except
, where its argument is zero, and where the function is infinite, as shown in Fig. 1, A. Exercise
values ​​at the points of the argument are ambiguous due to its turning to infinity, therefore the delta function is generalized function , and requires further definition in the form of normalization.

Fig.1. Delta function

Normalization condition

,
. (2.2)

The area under the graph of a function is equal to one in any interval containing a point a, as shown in Figure 1, b. Therefore, the delta function models a point disturbance of unit magnitude.

Function parity follows from (2.1)

,

. (2.2a)

From symmetry
relative to the point
we get

, (2.2b)

as follows from Fig. 1, b.

Orthonormality. Lots of features

,
,

forms an orthonormal infinite-dimensional basis.

The delta function was used in optics by Kirchhoff in 1882, and in electromagnetic theory by Heaviside in the 90s of the 19th century.

Gustav Kirchhoff (1824–1887) Oliver Heaviside (1850–1925)

Oliver Heaviside, a self-taught scientist, was the first to use vectors in physics, developed vector analysis, introduced the concept of an operator and developed operational calculus - an operator method for solving differential equations. He introduced the switching function, which was later named after him, and used a point impulse function - the delta function. Applied complex numbers in the theory of electrical circuits. For the first time he wrote down Maxwell's equations in the form of 4 equalities instead of 20 equations, as Maxwell had. Introduced terms: conductivity, impedance, inductance, electret . Developed the theory of telegraph communication over long distances, predicted the presence of an ionosphere near the Earth - Kennelly–Heaviside layer .

The mathematical theory of generalized functions was developed by Sergei Lvovich Sobolev in 1936. He was one of the founders of the Novosibirsk Academic Town. The Institute of Mathematics of the SB RAS is named after him, the founder and director of which he was from 1957 to 1983.

Sergei Lvovich Sobolev (1908–1989)

Properties of the delta function Filter property

For a smooth function
, which has no discontinuities, from (2.1)

we get filtering property of the delta function in differential form , affecting one point
:

We believe
, and use the limit for the delta function at
, shown in Fig. 1, b. We find

,

. (2.4)

Let's integrate (2.3) over the interval
, including the point a, we take into account normalization (2.2) and obtain filtering property of the delta function in integral form

,
. (2.5)

Orthonormality of the basis

In (2.5) we assume

,
,

and we obtain the condition for the basis to be orthonormal
with a continuous range of values

. (2.7)