| Summary: | The <i>q</i>-exponential form <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>e</mi><mi>q</mi><mi>x</mi></msubsup><mo>≡</mo><msup><mrow><mo>[</mo><mn>1</mn><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo>)</mo></mrow><mi>x</mi><mo>]</mo></mrow><mrow><mn>1</mn><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo>)</mo></mrow></msup><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mrow><mo>(</mo><msubsup><mi>e</mi><mn>1</mn><mi>x</mi></msubsup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is obtained by optimizing the nonadditive entropy <inline-formula><math display="inline"><semantics><mrow><msub><mi>S</mi><mi>q</mi></msub><mo>≡</mo><mi>k</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mo>∑</mo><mi>i</mi></msub><msubsup><mi>p</mi><mi>i</mi><mi>q</mi></msubsup></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></semantics></math></inline-formula> (with <inline-formula><math display="inline"><semantics><mrow><msub><mi>S</mi><mn>1</mn></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>B</mi><mi>G</mi></mrow></msub><mo>≡</mo><mo>−</mo><mi>k</mi><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><mo form="prefix">ln</mo><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from <i>q</i>-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding <i>q</i>-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing <inline-formula><math display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> as particular case.
|
|---|