Mathematics
"Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also." David Hilbert (1900) [1]
"Every mathematician knows it is impossible to understand an elementary course in thermodynamics." Vladimir I. Arnold (1989)[2]
Here are details of mathematical tools that have applications in classical thermodynamics, statistical mechanics and computer simulation techniques.
General
- Clebsch-Gordan coefficients
- Euler angles
- Gauss theorem
- Green's theorem
- Lagrange multipliers
- Laplace's equation
- Lee-Yang circle theorem
- Linear algebra
- Liouville's theorem
- Lyapunov exponents
- Padé approximants
- Perron-Frobenius theorem
- Poincaré theorem
- Power series
- Predictor-corrector methods
- Quaternions
- Random numbers
- Root finding
- Runge-Kutta method
- Spherical harmonics
- Stirling's approximation
- Substantive derivative
- Taylor expansion
- Trotter formula
- Wigner D-matrix
- Wigner rotation matrices
Functions and distributions
- Bessel functions
- Dirac delta distribution
- Gamma function
- Gaussian distribution
- Heaviside step distribution
- Kronecker delta
- Poisson distribution
- Ramp function
Geometry
Integrals and quadrature
- Standard integrals
- Elliptic integrals
- Newton-Cotes formulas
- Gauss-Chebyshev quadrature
- Gauss-Legendre quadrature
Polynomials
Transforms
Computer assisted mathematics
| Packages | Program or library | Notes | License |
|---|---|---|---|
| BLAS | routines | Basic Linear Algebra Subprograms | free source |
| CGAL | library | Computational Geometry Algorithms Library | free, Open Source |
| FreeMat | program | free | |
| IMSL Numerical Libraries | library | ||
| LAPACK | routines | Linear Algebra PACKage | free source |
| ScaLAPACK | library | Scalable LAPACK | free source |
| Magma | |||
| Maple | program | commercial | |
| Mathematica | program | commercial | |
| Math Kernel Library | library | ||
| MATLAB | program | commercial | |
| Maxima | program | free (General Public License) | |
| NAG | routines | Numerical Algorithms Group | commercial |
| Numerical Recipes | routines | ||
| Octave | program | free (GNU) | |
| QUADPACK | routines | free | |
| SAGE | program | free (General Public License) | |
| Scilab | program | free, Open Source |
Interesting reading
- Eugene P. Wigner "The unreasonable effectiveness of mathematics in the natural sciences", Communications on Pure and Applied Mathematics 13 pp. 1-14 (1960)
- David Ruelle "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the American Mathematical Society 19 pp. 259-268 (1988)
References
- ↑ David Hilbert, excerpt from problem VI of his lecture "Sur les problèmes futurs des Mathématiques", Second International Congress of Mathematicians, pp. 58-114 Paris (1900)
- ↑ Vladimir I. Arnold "Contact geometry: The geometrical method of Gibbs' thermodynamics," in Proceedings of the Gibbs Symposium, Yale University, May 15-17, 1989 Ed. D. G. Caldi and G. D. Mostow, American Mathematical Society (1990) ISBN 978-0-8218-0157-4 p. 163.