Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium
Non-equilibrium evolution at absolute temperature <i>T</i> and approach to equilibrium of statistical systems in long-time (<i>t</i>) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension,...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Dynamics |
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| Online Access: | https://www.mdpi.com/2673-8716/3/2/20 |
| _version_ | 1797595189990129664 |
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| author | Ramon F. Álvarez-Estrada |
| author_facet | Ramon F. Álvarez-Estrada |
| author_sort | Ramon F. Álvarez-Estrada |
| collection | DOAJ |
| description | Non-equilibrium evolution at absolute temperature <i>T</i> and approach to equilibrium of statistical systems in long-time (<i>t</i>) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>), is described by the non-equilibrium reversible Liouville distribution (<i>W</i>) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mrow><mi>e</mi><mi>q</mi></mrow></msub></semantics></math></inline-formula> generates orthogonal (Hermite) polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>n</mi></msub></semantics></math></inline-formula> in momenta. Suitable moments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-formula> of <i>W</i> (using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>n</mi></msub></semantics></math></inline-formula>’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-<i>t</i> approximation, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-formula>’s yield irreversibly approach to equilibrium. The approach is extended (without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>) to: (i) a non-equilibrium system of <i>N</i> classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> field theory (without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>). The extension to one non-relativistic quantum particle (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>) employs the non-equilibrium Wigner function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>Q</mi></msub></semantics></math></inline-formula>): difficulties related to non-positivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>Q</mi></msub></semantics></math></inline-formula> are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> field theory (a meson gas off-equilibrium, without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> theory at high <i>T</i> and large distances and long <i>t</i> occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> one, yielding an approach to equilibrium. |
| first_indexed | 2024-03-11T02:33:17Z |
| format | Article |
| id | doaj.art-0351943b85394751a41bbf83ec3781e0 |
| institution | Directory Open Access Journal |
| issn | 2673-8716 |
| language | English |
| last_indexed | 2024-03-11T02:33:17Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Dynamics |
| spelling | doaj.art-0351943b85394751a41bbf83ec3781e02023-11-18T10:04:42ZengMDPI AGDynamics2673-87162023-06-013234537810.3390/dynamics3020020Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-EquilibriumRamon F. Álvarez-Estrada0Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, SpainNon-equilibrium evolution at absolute temperature <i>T</i> and approach to equilibrium of statistical systems in long-time (<i>t</i>) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>), is described by the non-equilibrium reversible Liouville distribution (<i>W</i>) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mrow><mi>e</mi><mi>q</mi></mrow></msub></semantics></math></inline-formula> generates orthogonal (Hermite) polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>n</mi></msub></semantics></math></inline-formula> in momenta. Suitable moments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-formula> of <i>W</i> (using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>n</mi></msub></semantics></math></inline-formula>’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-<i>t</i> approximation, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-formula>’s yield irreversibly approach to equilibrium. The approach is extended (without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>) to: (i) a non-equilibrium system of <i>N</i> classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> field theory (without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>). The extension to one non-relativistic quantum particle (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>) employs the non-equilibrium Wigner function (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>Q</mi></msub></semantics></math></inline-formula>): difficulties related to non-positivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>Q</mi></msub></semantics></math></inline-formula> are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> field theory (a meson gas off-equilibrium, without <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mi>b</mi></mrow></semantics></math></inline-formula>), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> theory at high <i>T</i> and large distances and long <i>t</i> occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ϕ</mi><mn>4</mn></msup></semantics></math></inline-formula> one, yielding an approach to equilibrium.https://www.mdpi.com/2673-8716/3/2/20statistical systemsclassical particlesquantum fieldsapproach to equilibrium |
| spellingShingle | Ramon F. Álvarez-Estrada Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium Dynamics statistical systems classical particles quantum fields approach to equilibrium |
| title | Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium |
| title_full | Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium |
| title_fullStr | Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium |
| title_full_unstemmed | Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium |
| title_short | Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium |
| title_sort | approach to equilibrium of statistical systems classical particles and quantum fields off equilibrium |
| topic | statistical systems classical particles quantum fields approach to equilibrium |
| url | https://www.mdpi.com/2673-8716/3/2/20 |
| work_keys_str_mv | AT ramonfalvarezestrada approachtoequilibriumofstatisticalsystemsclassicalparticlesandquantumfieldsoffequilibrium |