Abstract
Various physical processes and phenomena are described by partial differential equations (PDEs).
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Notes
- 1.
Commonly referred to as emergent.
- 2.
Gaspard Monge (1746–1818).
- 3.
Jean-Baptiste le Rond d’Alembert (1717–1783).
- 4.
Jean-Marie Constant Duhamel (1797–1872).
- 5.
The mean kinetic energy satisfies \(\frac{1}{2}m\left\langle \mathbf{v}^2\right\rangle =\frac{1}{2}\left\langle v_x^2+v_y^2+v_z^2\right\rangle =\frac{3}{2}k_BT\), where \(\left\langle v_x^2\right\rangle =\left\langle v_y^2\right\rangle =\left\langle v_z^2\right\rangle =k_BT/m\) for an isotropic thermal distribution at temperature T.
- 6.
\(\lambda \) reduces by \(\sqrt{2}\) in a thermal distribution, see [2].
- 7.
Jacobus H. van ’t Hoff (1852–1911).
- 8.
Fick’s second law with convection.
- 9.
For a related discussion, see [4].
- 10.
Here, the factor of one-half can be seen to derive from integration of the binding energy \(dU=-(GM/R)dM\) in case of a constant radius R.
- 11.
A more refined analysis takes into account a density in the core that is about 58 times the mean density, showing a significant increase in the diffusion time; see [5].
- 12.
See, e.g., [8].
- 13.
Conserved quantities along characteristic directions are Riemann invariants, that allow analytic solutions to, e.g., the nonlinear equations of compressible gas dynamics.
References
Garabedian, P.R., Partial Differential Equations, Chelsia Publishing, New York (1986).
S. Chapman and T.G. Cowling, The Mathematical theory of non-uniform gases (Cambridge University Press, 1990).
Fick, A.E., 1855, Über Diffusion, Annal. Physik und Chemie, 94, 59.
Pekir, G., 2003, Stoch. Models, 19, 383.
Mitalas, R., & Sills, K.R., 1992, ApJ, 401, 759.
Hadamard, J.S., 1923, Lectures on Cauchy’s problem in linear differential equations (Yale University Press, 1923; Reprint Dover 2003).
van Putten, M.H.P.M., 1994, Phys. Rev. D, 50, 6640.
van Putten, M.H.P.M., 2006, PNAS, 103, 519.
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van Putten, M.H. (2017). Linear Partial Differential Equations. In: Introduction to Methods of Approximation in Physics and Astronomy. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-10-2932-5_4
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