Stochastic dynamics and Boltzmann hierarchy:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Berlin [u.a.]
de Gruyter
2009
|
| Series: | De Gruyter expositions in mathematics
48 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIII, 295 S. 25 cm |
| ISBN: | 9783110208047 |
Staff View
MARC
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| 245 | 1 | 0 | |a Stochastic dynamics and Boltzmann hierarchy |c by D. Ya. Petrina |
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Record in the Search Index
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| adam_text |
CONTENTS INTRODUCTION 1 1 SYSTEM OF HARD SPHERES 14 1.1 INTRODUCTION 14
1.2 HAMILTONIAN DYNAMICS OF A SYSTEM OF HARD SPHERES 14 1.2.1 HAMILTON
EQUATIONS 14 1.2.2 EXISTENCE OF TRAJECTORIES 16 1.2.3 LIOUVILLE THEOREM
18 1.3 EVOLUTION OPERATOR FOR A SYSTEM OF HARD SPHERES 18 1.3.1
DEFINITION OF THE EVOLUTION OPERATOR 18 1.3.2 LIOUVILLE EQUATION 20
1.3.3 EVOLUTION OPERATOR AND LIOUVILLE EQUATION FOR NEGATIVE TIME 23 1.4
BBGKY HIERARCHY FOR SYSTEMS OF HARD SPHERES 24 1.4.1 DEFINITION OF
CORRELATION FUNCTIONS 24 1.4.2 DERIVATION OF HIERARCHY OF EQUATIONS FOR
CORRELATION FUNCTIONS 27 1.4.3 SOLUTION OF THE BBGKY HIERARCHY IN THE
SPACE OF SUMMABLE FUNCTIONS 28 1.4.4 SOLUTION OF THE BBGKY HIERARCHY 30
1.4.5 BBGKY HIERARCHY WITH NONSTANDARD NORMALIZATION 32 1.5
JUSTIFICATION OF THE BOLTZMANN-GRAD LIMIT 34 1.5.1 DEFINITION OF THE
BOLTZMANN-GRAD LIMIT 34 1.5.2 AUXILIARY LEMMAS 36 1.5.3 CONVERGENCE OF
SOLUTIONS OF THE BBGKY HIERARCHY OF A SYS- TEM OF HARD SPHERES TO
SOLUTIONS OF THE ORDINARY BOLTZMANN HIERARCHY IN THE BOLTZMANN-GRAD
LIMIT 38 1.5.4 CONVERGENCE OF SOLUTIONS OF THE BBGKY HIERARCHY OF SYS-
TEMS OF HARD SPHERES TO SOLUTIONS OF THE PROPER STOCHASTIC HIERARCHY IN
THE BOLTZMANN-GRAD LIMIT 41 2 STOCHASTIC DYNAMICS AS THE LIMIT OF THE
HAMILTONIAN DYNAMICS OF HARD SPHERES 44 2. BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/991613503 DIGITALISIERT DURCH CONTENTS 2.2 STOCHASTIC
TRAJECTORIES AS THE LIMIT OF THE HAMILTONIAN TRAJECTORIES OF HARD
SPHERES 45 2.2.1 HAMILTONIAN TRAJECTORIES OF HARD SPHERES 45 2.2.2
STOCHASTIC TRAJECTORIES 46 2.2.3 CONVERGENCE OF HAMILTONIAN TRAJECTORIES
TO STOCHASTIC TRA- JECTORIES 49 2.3 NEW REPRESENTATION OF HAMILTONIAN
AND STOCHASTIC TRAJECTORIES . 51 2.3.1 REPRESENTATION OF HAMILTONIAN
TRAJECTORIES 51 2.3.2 REPRESENTATION OF STOCHASTIC TRAJECTORIES 52 2.4
FUNCTIONAL FOR A SYSTEM OF TWO HARD SPHERES 55 2.4.1 DOMAIN OF
INTERACTION AND FUNCTIONAL 55 2.4.2 DERIVATIVE OF FUNCTIONAL 58 2.5
FUNCTIONAL FOR A SYSTEM OF TWO STOCHASTIC PARTICLES 60 2.5.1 FUNCTIONAL
OF STOCHASTIC PARTICLES AS THE LIMIT OF THE FUNC- TIONAL OF HARD SPHERES
60 2.5.2 DERIVATIVE OF FUNCTIONAL WITH RESPECT TO TIME 64 2.6 GENERAL
CASE OF MANY-PARTICLE SYSTEM 67 2.6.1 FUNCTIONAL FOR MANY HARD SPHERES
67 2.6.2 DERIVATIVE OF FUNCTIONAL WITH RESPECT TO TIME 69 2.6.3 LIMIT OF
THE AVERAGE OF THE FUNCTIONAL FOR HARD SPHERES AND THE FUNCTIONAL OF
STOCHASTIC PARTICLES 70 2.7 INFINITESIMAL OPERATOR OF THE EVOLUTION
OPERATOR OF STOCHASTIC PARTICLES 76 2.7.1 DYNAMICS OF FINITELY MANY
PARTICLES 76 2.7.2 EVOLUTION OPERATOR OF FINITELY MANY PARTICLES AND ITS
IN- FINITESIMAL OPERATOR 79 2.7.3 EVOLUTION OPERATOR FOR NEGATIVE TIME
84 2.7.4 EQUIVALENCE OF THE INFINITESIMAL OPERATORS 89 STOCHASTIC
BOLTZMANN HIERARCHY 93 3. CONTENTS XI 3.4.2 DERIVATION OF THE STOCHASTIC
BOLTZMANN HIERARCHY FROM THE ITO-LIOUVILLE EQUATION 122 3.4.3 DERIVATION
OF ORDINARY BOLTZMANN HIERARCHY FROM BOLTZ- MANN EQUATION 125 3.5
DERIVATION OF STOCHASTIC BOLTZMANN HIERARCHY FROM BBGKY HIERAR- CHY FOR
HARD SPHERES 126 3.5.1 STOCHASTIC BOLTZMANN HIERARCHY 126 3.5.2
SOLUTIONS OF THE ORDINARY BOLTZMANN HIERARCHY AND THE BOLTZMANN-GRAD
LIMIT OF SOLUTIONS OF THE BBGKY HIERARCHY 129 3.5.3 DERIVATION OF THE
STOCHASTIC BOLTZMANN HIERARCHY FROM THE EVOLUTION OPERATOR OF THE BBGKY
HIERARCHY FOR HARD SPHERES 131 3.5.4 FUNCTIONAL FOR CORRELATION
FUNCTIONS 134 3.5.5 STOCHASTIC BOLTZMANN HIERARCHY 136 3.5.6 DIFFERENT
REPRESENTATIONS OF THE INFINITESIMAL OPERATOR . . . 139 3.5.7 DIFFERENT
EQUIVALENT FORMS OF THE STOCHASTIC BOLTZMANN HI- ERARCHY 141 3.6
BOLTZMANN EQUATION AND ITS SOLUTIONS IN TERMS OF STOCHASTIC DYNAMICS 142
3.6.1 ITERATIONS OF THE BOLTZMANN EQUATION 142 3.6.2 ITERATIONS OF THE
BOLTZMANN HIERARCHIES 146 SOLUTIONS OF THE STOCHASTIC BOLTZMANN
HIERARCHY 149 4.1 INTRODUCTION 149 4.2 SOLUTIONS OF THE STOCHASTIC
HIERARCHY IN THE SPACE OF BOUNDED FUNCTIONS 150 4.2.1 ABSTRACT FORM OF
THE STOCHASTIC HIERARCHY 150 4.2.2 CONVERGENCE OF SERIES (4.2.8) IN THE
SPACE $,/} 152 4.2.3 ONE AUXILIARY LEMMA 154 4.2.4 CONVERGENCE OF
SERIES (4.2.8) IN THE SPACE EC Y SS 156 4. XII CONTENTS 5.3.2 HIERARCHY
WITH FIXED RANDOM VECTORS 178 5.4 REPRESENTATION OF SOLUTIONS OF THE
SPATIALLY HOMOGENEOUS HIERARCHY . 179 5.4.1 REPRESENTATION OF SOLUTIONS
OF THE SPATIALLY HOMOGENEOUS HIERARCHY THROUGH SERIES OF ITERATIONS 179
5.4.2 ONE-PARTICLE DISTRIBUTION FUNCTION AS A SOLUTION OF THE BOLTZ-
MANN EQUATION 185 6 STOCHASTIC DYNAMICS FOR THE BOLTZMANN EQUATION WITH
ARBITRARY DIFFEREN- TIAL SCATTERING CROSS SECTION 189 6.1 INTRODUCTION
189 6.2 STOCHASTIC DYNAMICS 191 6.2.1 FUNCTIONAL AVERAGE 191 6.2.2
INFINITESIMAL OPERATOR WITH FIXED RANDOM VECTORS 198 6.2.3 DUALITY
PRINCIPLE 200 6.3 HIERARCHY FOR CORRELATION FUNCTIONS 205 6.3.1
DERIVATION OF HIERARCHY FROM EQUATION FOR DISTRIBUTION FUNCTION205 6.3.2
DERIVATION OF HIERARCHY FROM FUNCTIONAL AVERAGE 209 6.4 SOLUTIONS OF THE
STOCHASTIC HIERARCHY 213 6.4.1 ABSTRACT FORM OF THE STOCHASTIC HIERARCHY
213 6.4.2 CHAOS PROPERTY 214 6.4.3 SPATIALLY HOMOGENEOUS INITIAL DATA
217 6.5 STOCHASTIC PROCESS IN MOMENTUM SPACE 218 6.5.1 AVERAGING
PROCEDURE IN SPATIALLY HOMOGENEOUS CASE . 218 6.5.2 DIFFERENTIAL
EQUATION FOR SPATIALLY HOMOGENEOUS DISTRIBUTION FUNCTIONS 220 6.5.3
HIERARCHY FOR CORRELATION FUNCTIONS IN MEAN-FIELD APPROXI- MATION 220 7
ANALOG OF LIOUVILLE EQUATION AND BBGKY HIERARCHY FOR A SYSTEM OF HARD
SPHERES WITH INELASTIC COLLISIONS 223 7.1 INTRODUCTION 223 7. CONTENTS
XIII 7.4.4 GRAND CANONICAL ENSEMBLE 247 APPENDIX A 248 APPENDIX B 250 8
BBGKY HIERARCHY SOLUTION FOR A HARD SPHERES SYSTEM WITH INELASTIC COLLI-
SIONS 252 8.1 INTRODUCTION 252 8.2 SOLUTION OF HIERARCHY FOR CORRELATION
FUNCTIONS 254 8.2.1 SOLUTION FORMULA 254 8.2.2 CONVERGENCE OF SERIES 257
8.2.3 GROUP PROPERTY 258 8.2.4 STRONG CONTINUITY OF THE GROUP 259 8.3
INFINITESIMAL GENERATOR OF THE GROUP AND A SOLUTION OF THE BBGKY
HIERARCHY 261 8.3.1 INFINITESIMAL GENERATOR 261 8.3.2 EXISTENCE OF
SOLUTIONS OF THE BBGKY HIERARCHY 262 8.3.3 STATES OF INFINITE SYSTEMS
263 8.4 STOCHASTIC BOLTZMANN HIERARCHY FOR GRANULAR FLOW 264 8.4.1
STOCHASTIC DYNAMICS FOR HARD SPHERES WITH INELASTIC COLLISIONS 264 8.4.2
STOCHASTIC TRAJECTORIES AND OPERATOR OF EVOLUTION 264 8.4.3 FUNCTIONAL
AVERAGE 265 8.4.4 HIERARCHY FOR CORRELATION FUNCTIONS 267 8.4.5 SOLUTION
OF THE STOCHASTIC BOLTZMANN HIERARCHY 268 8.4.6 ORDINARY BOLTZMANN
HIERARCHY 270 REFERENCES 273 INDEX 296 |
| any_adam_object | 1 |
| author | Petrina, Dmytro Jakovyč 1934-2006 |
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| author_facet | Petrina, Dmytro Jakovyč 1934-2006 |
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| author_sort | Petrina, Dmytro Jakovyč 1934-2006 |
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| dewey-full | 530.13 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.13 |
| dewey-search | 530.13 |
| dewey-sort | 3530.13 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035759192 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-20T10:21:02Z |
| institution | BVB |
| isbn | 9783110208047 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018619096 |
| oclc_num | 434507992 |
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| owner | DE-634 DE-824 DE-384 DE-11 DE-83 |
| owner_facet | DE-634 DE-824 DE-384 DE-11 DE-83 |
| physical | XIII, 295 S. 25 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | de Gruyter |
| record_format | marc |
| series | De Gruyter expositions in mathematics |
| series2 | De Gruyter expositions in mathematics |
| spelling | Petrina, Dmytro Jakovyč 1934-2006 Verfasser (DE-588)138912378 aut Stochastic dynamics and Boltzmann hierarchy by D. Ya. Petrina Berlin [u.a.] de Gruyter 2009 XIII, 295 S. 25 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter expositions in mathematics 48 Boltzmann-Gleichung - Stochastischer Prozess Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Boltzmann-Gleichung (DE-588)4146261-0 gnd rswk-swf Boltzmann-Gleichung (DE-588)4146261-0 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 De Gruyter expositions in mathematics 48 (DE-604)BV004069300 48 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3189276&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018619096&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Petrina, Dmytro Jakovyč 1934-2006 Stochastic dynamics and Boltzmann hierarchy De Gruyter expositions in mathematics Boltzmann-Gleichung - Stochastischer Prozess Stochastischer Prozess (DE-588)4057630-9 gnd Boltzmann-Gleichung (DE-588)4146261-0 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4146261-0 |
| title | Stochastic dynamics and Boltzmann hierarchy |
| title_auth | Stochastic dynamics and Boltzmann hierarchy |
| title_exact_search | Stochastic dynamics and Boltzmann hierarchy |
| title_full | Stochastic dynamics and Boltzmann hierarchy by D. Ya. Petrina |
| title_fullStr | Stochastic dynamics and Boltzmann hierarchy by D. Ya. Petrina |
| title_full_unstemmed | Stochastic dynamics and Boltzmann hierarchy by D. Ya. Petrina |
| title_short | Stochastic dynamics and Boltzmann hierarchy |
| title_sort | stochastic dynamics and boltzmann hierarchy |
| topic | Boltzmann-Gleichung - Stochastischer Prozess Stochastischer Prozess (DE-588)4057630-9 gnd Boltzmann-Gleichung (DE-588)4146261-0 gnd |
| topic_facet | Boltzmann-Gleichung - Stochastischer Prozess Stochastischer Prozess Boltzmann-Gleichung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3189276&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018619096&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004069300 |
| work_keys_str_mv | AT petrinadmytrojakovyc stochasticdynamicsandboltzmannhierarchy |