| Summary: | 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.
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