Contents

MINISTRY OF EDUCATION AND SCIENCE OF RUSSIAN FEDERATION

[1mm] SAMARA STATE UNIVERSITY


MATHEMATICAL PHYSICS

[5mm] PROBLEMS AND SOLUTIONS

[5mm]



The Students Training Contest Olympiad

in Mathematical and Theoretical Physics

(on May 21st – 24th, 2010)

[26mm]

Special Issue № 3 of the Series

[1mm] «Modern Problems of Mathematical Physics»

Samara

Samara University Press

2010

УДК 51-7+517.958

ББК 22.311
М 34

Authors:
G.S. Beloglazov, A.L. Bobrick, S.V. Chervon, B.V. Danilyuk, M.V. Dolgopolov,
M.G. Ivanov, O.G. Panina, E.Yu. Petrova, I.N. Rodionova, E.N. Rykova,
M.Y. Shalaginov, I.S. Tsirova, I.V. Volovich, A.P. Zubarev

М 34    y Mathematical Physics : Problems and Solutions of The Students Training Contest Olympiad in Mathematical and Theoretical Physics (May 21st – 24th, 2010) / [G.S. Beloglazov et al.]. – Ser. «Modern Problems of Mathematical Physics». – Spec. Iss. № 3. – Samara : Samara University Press, 2010. – 68 p.: il.

ISBN 978-5-86465-494-1

The present issue of the series <<Modern Problems in Mathematical Physics>> represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia’s Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation.

The subjects covered by the problems include classical mechanics, integrable nonlinear systems, probability, integral equations, PDE, quantum and particle physics, cosmology, and other areas of mathematical and theoretical physics.

The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics.

УДК 51-7+517.958

ББК 22.311  yyyyyy

Editors: B.V. Danilyuk, M.V. Dolgopolov, M.G. Ivanov, I.S. Tsirova, I.V. Volovich

Invited reviewers:   M. N. Dubinin, Skobeltsyn Institute of Nuclear Physics of Moscow State University,   and   Yu. N. Radayev, Institute for Problems in Mechanics of the Russian Academy of Sciences

The Olympiad and the given edition are supported by the grants

ADTP №  3341, 10854 and FTP № 5163

of the Ministry of Education and Science of the Russian Federation,

and by Training and retrainings of specialist center

of Samara State University.

[3mm] Information support on the website   www.labmathphys.samsu.ru/eng

ISBN 978-5-86465-494-1                   © Authors, 2010

© Samara State University, 2010

© Scientific Research Laboratory of Mathematical Physics, 2010

© Registration. Samara University Press, 2010

Introduction

 

1   Regulations on The Olympiad

Regulations on holding The Olympiad contest for students on Mathematical and Theoretical Physics were developed in April 2010 [see Special Issue No. 2]. They were signed by the three parties: Samara State University (hereinafter referred to as SamGU), Steklov Mathematical Institute (SMI RAS), and Moscow Institute of Physics and Technology (MIPT). The text of the regulations is given below.

2   Carrying out The Olympiad

On May 21-24th, 2010, All-Russian Student Training Olympiad in Mathematical and Theoretical Physics "Mathematical Physics" with International Participation has been held. It was the second in the series of Olympiads. It is planned that in future such Olympiads will take place annually.

The organizers of the series of Olympiads on Mathematical & Theoretical Physics "Mathematical Physics" are:

Aleksander Anatolyevitch Andreyev (staff member of the Scientific Research Laboratory of Mathematical Physics of SamGU),

Georgiy Sergeyevitch Beloglazov (The University of Dodoma - UDOM, Tanzania; Perm State Pharmaceutical Academy),

Boris Vasilyevitch Danilyuk (staff member of the Scientific Research Laboratory of Mathematical Physics of SamGU),

Mikhail Vyacheslavovitch Dolgopolov (Head of the Scientific Research Laboratory of Mathematical Physics of SamGU),

Vitaliy Petrovitch Garkin (Vice-rector for Academic Affairs, Chairman of the Local Organizing Committee, SamGU),

Mikhail Gennadievich Ivanov (Associate Professor, MIPT),

Yuri Nikolayevitch Radayev (staff member of the Scientific Research Laboratory of Mathematical Physics of SamGU),

Irina Nikolayevna Rodionova (staff member of the Scientific Research Laboratory of Mathematical Physics of SamGU),

Yuri Aleksandrovitch Samarskiy (Deputy Vice Chancellor on Education, MIPT),

Irina Semyonovna Tsirova (docent, SamGU),

Igor Vasilyevitch Volovich (scientific leader of the Scientific Research Laboratory of Mathematical Physics of SamGU, head of the department of Mathematical Physics of MIAN),

Aleksander Petrovitch Zubarev (staff member of the Scientific Research Laboratory of Mathematical Physics of SamGU).

The Olympiad has been held as a team competition. Number of participants of each team – from 3 to 10 students of 2nd to 6th courses (years) of higher educational establishments of Russia, CIS, and other countries. It was allowed that more than one team participates on behalf of any organization. Order of the Olympiad:

The participants have been offered to solve 14 problems. Time to start solving problems of the contest was 11:00 pm Moscow time on May 20th, 2010. The statements of the contest tasks are published in *.pdf format at the webpage of the Olympiad www.labmathphys.samsu.ru/eng/content/view/29/36/ of the website of the Scientific Research Laboratory of Mathematical Physics of SamGU

www.labmathphys.samsu.ru/eng

and have been sent to the registered participants of The Olympiad.

The deadline to send the scanned (or photographed) solutions to the E-mail address of the Mathematical Physics Laboratory: slmp@ssu.samara.ru was 11 pm Moscow time on May 24th, 2010. All participants of The Olympiad who had sent their solutions by E-mail, have received confirmation that their solutions had been accepted.

It was allowed that the participants solve any problems from the number of the proposed ones which they find affordable for the own level of knowledge digestion in different units of mathematics and physics thus participating in the topical scoring nomination (for purpose of this scoring nomination, the problems are aggregated into groups 1 to 3 problems in each).

In the application letter, the name of organization hosting the team should be stated together with the surname, name, for each participant of the team, Department (speciality), course/year; contact E-mail address.

The Nominations of The Olympiad:

1) The overall team scoring based on the three best team participants performance (3 prize-winning team places). In the present Olympiad, it is possible to submit only one solution on behalf of a team; it is advised to mention the author(s) of every solution or solution method [stating also the year(s)/course(s) of studying] at the end of each solution (or method of solution). The winner is the team which participants have solved correctly maximum number of different problems. Any participant of a team has the right to send a solution separately. Within the team scoring, the correct solutions will be considered and accounted. The maximum possible number of points in a team scoring is 14 (because the total number of problems offered is 14).

2) It is possible for a student to participate in the overall personal contest (within the framework of the Olympiad by correspondence) ON CONDITION OF THE PRESENCE OF AN INDIVIDUAL APPLICATION (REQUEST) from a participant of The Olympiad (3 prize-winning places).

3) Overall team topic scoring (1 – 3 prize-winning places on each subject).

4) Separate team scoring among each of the years (second through sixth courses).

5) Best team among the technical specialities of the institutes of higher education.

6) Other nominations. Separate nomination is supported by the Center on Advanced Training and Professional Development at Samara State University.

The winners of The Olympiad held by correspondence participated in the day competition Olympiad held in Samara in September - 2010 (at the same time with the Second International Conference and School on Mathematical Physics and its Applications). For the above said winners, their travel and/or accommodation expenses were reimbursed.

3   Contents of the problems for the Olympiad contest

The topic range of our ’Olympiad’ is related to mathematical methods in describing physical phenomena based on the following units of mathematics and theoretical physics:

theory of differential, integral equations, and boundary-value problems;

theory of generalized functions, integral transform, theory of functions of complex variable;

functional analysis, operational calculus, spectral analysis;

probability theory, theory of random processes;

differential geometry and topology;

theoretical mechanics, electrodynamics, relativity theory, quantum mechanics, and gravitation theory.

New scientific methodological approach to composing the statements of the problems for The Olympiad was first introduced in the sense that about a half of the problems offered to the participants for the solution supposed that certain stage of research (taken from original modern academic research in mathematical physics and its applications) is involved. On the basis of the above mentioned approach, the recommendations on composing statements of the problems for The Olympiad are developed.

In the present issue, we quote the statements of problems offered to the participants of All-Russia Students Training Olympiad in Mathematical and Theoretical Physics "Mathematical Physics" with International Participation (held on May 21-24th, 2010).

4   Results and resume of The Olympiad

In The Olympiad, the teams from the following institutes of higher education and other organizations have participated:

Belarusian State university,

Moscow Institute of Physics and Technology (State University),

National University of Singapore,

Department of Theor. Phys. named after I.E.Tamm of FIAN (the Institute of Physics of Academy of Sciences of Russia) - postgraduate,

Samara State University of Architecture and Construction (two teams),

Samara State Aerospace University (SGAU),

Samara State University,

The Federal University of Siberia,

Ulyanovsk State Pedagogical University, UlGPU (two teams),

The University of Dodoma (UDOM, Tanzania),

Yaroslavl State University (the team of the Physics Department).

The jury has positively assessed the works by the following participants of the teams:

Belarusian State University: Alexey Bobrick.

Moscow Institute of Physics and Technology (State University): Kostjukevich Yury, and the fourth course team: Nikolai Fedotov, Anton Fetisov, Mikhail Shalaginov, Aleksander Shtyk.

Department of Theor. Phys. named after I.E.Tamm of FIAN (the Institute of Physics of Academy of Sciences of Russia): Andrey Borisov.

Samara State University of Architecture and Construction, SGASU (two teams of the students of the 5th year): leader – Sergey Zinakov.

Samara State Aerospace University: Mikhail Malyshev, Yekaterina Pudikova.

Samara State University: team of theoretical physicists – Tatiana Volkova, Matvei Mashchenko, Maksim Nefedov, Yelena Petrova.

The Federal University of Siberia: Artyom Ryasik, Polina Syomina, Anton Sheykin.

Ulyanovsk State Pedagogical University (two teams): 3rd year – Yuri Antonov, Aleksandra Volkova, Oksana Rodionova;

5th year: Maria Vasina, Artyom Ovchinnikov. Aleksander Chaadayev, Aleksander Ernezaks.

The diploma of Laureates or diploma of the winners in nominations have been sent to all above mentioned participants. All participants of The Olympiad have been invited to attend the School-2010 on Applied Mathematical Physics (PMF) from July 1st till July 14th, and the scientific Conference together with another School & Olympiad (August 29th - September 9th, 2010).

The Winners of The Olympiad in the nominations:

The Overall Team Score:

1st Place, 5 problems solved correctly (means, 8 and more points per a problem, maximum 10 points per a problem), the total score is 102 points, - the team of the 4th year of MIPT. The winners are granted the prize - traveling costs be paid for them to participate in the scientific Conference together with School & Olympiad (August 29th - September 7th, 2010).

2nd Place - The Federal University of Siberia

3rd Place - Samara State University

The Total Personal Score:

1st Place, 4 problems solved correctly, score is 126 points, – Alexey Bobrick, Theoretical Physics magistracy at the Department of Physics of Belarusian State University. The winner is granted the prize - either traveling or accommodation costs be paid for him to participate in the scientific Conference together with School & Olympiad (August 29th - September 7th, 2010).

2nd Place, 3 problems solved correctly, score is 74 points, – Yuri Kostyukevitch, the student of the 5th year of the Department of Molecular and Biological Physics, group No. 541, MIPT. Recommended for the Magistracy or (post)graduate school of MIPT.

3rd Place, 2 problems solved correctly, score is 70 points, – Anton Sheykin, the student of the 4th year at Engineering Physical Department of IIFiRE (Physics and Radioelectronics) of Siberian Federal University. Recommended for the Magistracy or (post)graduate school of MIPT.

The best (complete and original) solutions of separate problems: by Alexey Bobrik, Polina Syomina, Mikhail Shalaginov, Anton Sheykin.

1st Team Place among the 3rd year students – SamGU;

2nd Team Place among the 3rd year students – SGAU.

1st Place in Personal contest among the 3rd year students – Maksim Nefedov;

2nd Place in Personal contest among the students of the 3rd year students – Mikhail Malyshev.

1st Place in Personal contest among the 4th year students – Anton Sheykin;

2 – 3 Places in Personal contest among the students of the 4th year – Nikolai Fedotov, Anton Fetisov, and Mikhail Shalaginov.

Among the teams of Pedagogical, Engineering & Technical institutes of higher education:

1st Place – UlGPU, 3 year;

2nd Place – UlGPU, 5 year;

3rd Place – SGASU.

All winners and prize winners of The Olympiad are granted with the free of charge accommodation at the PMF School-2010.

The information on the Olympiad, formulation of the Problems-2010 statements, answers and solutions of the tasks-2010 are presented in this document.

www.labmathphys.samsu.ru/eng            slmp@ssu.samara.ru

Organizers of a series of the Mathematical Physics Olympiads: Alexander Andreev, George Beloglazov, Boris Danilyuk, Mikhail Dolgopolov, Vitaliy Garkin, Mikhail Ivanov, Yury Radaev, Irina Rodionova, Yury Samarsky, Irina Tsirova, Igor Volovich, Alexander Zubarev

Problems and Solutions

 

Statements of the Problems

and Solutions at Students Training Olympiad

on Mathematical & Theoretical Physics

MATHEMATICAL PHYSICS

by Correspondence

with International Participation

May 21st – 24th, 2010

[4mm]

1. Virial for anharmonic oscillations

For a particle moving along the axis with Hamiltonian

where is a positive constant, is the mass of the particle, is the momentum of the particle, obtain the relationship between the average values of kinetic  and potential energy using two methods:
(a) directly from the virial theorem (see explanation below);
(b) from the condition

which is true due to the fact that the motion of the particle is finite.

Instruction: when a particle moves in a potential field, its Hamiltonian and acting force are defined by

Explanation to Problem 1. In classical mechanics time average values of kinetic and potential energies of the systems performing finite motion are in rather simple relationship.

The average value for a physical quantity for a sufficiently large time interval is defined in a standard way:

If is the average (for a rather long time interval) kinetic energy of the system of point particles (radius-vectors of the particles given as ) subjected to forces , then the following relation takes place:

The right hand side of equation () is called Clausius virial, and the equation itself expresses the so called the virial Theorem. The proof of the theorem is given, for example, in [1].

SOLUTION

(a) According to the given statement the particle is moving in the field of a potential force and possesses potential energy The force equals

Substitute it into the equation () which expresses the virial theorem:

(b) According to the given conditions,

that is why

2. Method of successive approximations

Solve the integral Volterra equation of 2nd kind

(1)

where , is a given (known) continuous on function, (continuously differentiable function), and , is the derivative. Perform your solution check.
Vito Volterra   (3 May 1860 –- 11 October 1940)   was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations.            

SOLUTION

Using method of successive approximations, we find the solution of the equation (\@setrefen2-z1 through kernel resolvent :

(2)
(3)
(4)
(5)

Using formula (\@setref25e we find the repeated kernel ,

(6)
(7)

Similarly, using the formula (\@setref25e,

(8)

we come to the conclusion that

(9)

Expression (\@setref29e should be substituted into the formula (\@setref23e:

(10)

Performing the transformation in (\@setref210 and recalling the expansion

(11)

we obtain

(12)

Substituting (\@setref212 into formula (\@setref22e, we obtain

(13)

Checking. Let us show that the function (\@setref213 is the solution of eq. (\@setrefen2-z1. Designate

(14)

and substitute the function (\@setref213 into the right side of equation (\@setref214. As a result, we shall obtain

(15)

In the last term of formula (\@setref215 let us change the integration order and calculate the inner integral:

(16)

The result should be substituted into the formula (\@setref215:

(17)

Our checking has shown that the function (\@setref213 is the correct solution of the equation (\@setrefen2-z1.

3. Evaluation for ultrametric diffusion

When solving equations of the ultrametric diffusion type (that have a relation to the description of conformational dynamics of complicated systems such as biomacromolecules) the results can often be presented in the form of series of exponents. Two of such series are represented below:

Here is time, and are probabilities that a system is in some definite groups of states, is some integer number, , are some parameters.

Study the asymptotic behavior of functions and at and evaluate, if possible, their asymptotics using elementary functions depending on .

SOLUTION

Let us explore and . Note that the function decreases, while the function increases with the growth of . Then in the interval the inequality takes place

takes place. Integrating it with respect to from to gives (for ):

Now, by summing over from to , we obtain:

where ,

By switching to new variables, we have:

Let us designate the function

considering for convergence.

Note that the limit for this function is the Gamma-function (see the proof in [2]):

Then at it is possible to write

Notation means that in the limit at the value of tends to zero.

Since

the final asymptotic evaluation is of the form :

Because

we have also

4. Double effort

Solve the Volterra integral equation

(1)

SOLUTION

The considered equation is in fact Volterra integral equation of kind with continuous kernel. According to the theory of linear integral equations, it has a unique solution. To find its solution, we reduce it to Cauchy problem for ordinary differential equation. 111Special Issue No. 4 contains also another methods of solution (see [2]).

Let us assume that is the solution of the equation (\@setref4e00. Double differentiation of the identity equation (\@setref4e00 by gives:

(2)
(3)
(4)

Assuming  results in:

(5)

At we have from eq. (\@setrefe11:

(6)

Hence , , and

(7)

It is possible to perform a check:

(8)
(9)

It is possible to propose other ways to solve this equation, such as with the help of Laplace transformation, or by building a resolvent of the kernel by successive approximations method. However, the technique developed above is the simplest.

5. Random walk

A particle performs random walk on one-dimensional lattice situated on the  axis, the nodes of the lattice have the coordinates . At the initial time moment the particle is at the origin of the coordinates. At random time moments , , ,…the particle performs the jumps into adjacent lattice nodes with the probabilities of a jump leftwards and rightwards equal to , the probability of remaining still being . The time intervals between the jumps , are independent random quantities which have the same exponential distribution with expectation . Find:

(a) dispersion of the location of the particle as time function ;

(b) probability that the particle is in -th node at time moment .

SOLUTION

1. Let us find the probability that the particle during time interval would perform exactly jumps, taking into account that this probability is the distribution function of the Poisson process. Let be time instants of the jumps, so that

(1)

Probability that the particle during the time interval would perform exactly jumps, can be represented as

(2)

where is expectation and

(3)

Let us perform Laplace transformation of the function :

(4)

Due to the fact that is the sum of independent random variables, then

(5)

where is Laplace image of . From this,

(6)

Due to the fact that , it follows that

(7)

and

(8)

Transforming from Laplace image to the original, we obtain the distribution function for the Poisson process

(9)

2. Let us find the dispersion of the location of the particle as a function of time . Location of the particle after jumps is defined by the random variable , where , are independent random variables, possessing the values with the probability and 0 with the probability . Due to the fact that , and for , the dispersion equals to

(10)

Substituting into this formula the expression for , we obtain

(11)

3. Let us find the probability that the particle is located at node at time moment . Let designate the probability that the particle is located at the point with coordinate after jumps (transitions). Then the probability that the particle is at location at time moment will be given by formula

(12)

Function equals to

(13)

where is the Kronecker delta. Due to the fact that , it is possible to write

(14)

Note that

(15)

Therefore

(16)

Substituting this equation for , and earlier found expression for into the formula for , we obtain

(17)