# Detailed balance: What does one rigourously mean by the transition rate $dw$?

Physics Asked by catalogue_number on August 5, 2020

Consider (for example) non-relativistic electron-impact ionisation, in which an effectively stationary atom in state $$I$$ is struck by an electron of momentum $${p_1}$$, producing an atom in state $$J$$ with two additional electrons, momenta $${p_2}, {p’_2}$$. The electron-impact ionisation transition rate is then expressed as $$dw(final|initial) = dw(p_2, p_2′; B| p_1; A)$$

The total rate is then given, in the presence of an electron distribution $$f(p)$$, as
$$Gamma_{eii} = N_e n_A int d^3p_1 f(p_1) dw(p_2, p_2′; B| p_1; A)$$
where $$N_e$$ is electron population and $$n_A$$ the number density of atoms in state $$A$$. Here, the interpretation $$dw = frac{p_1}{m_e}sigma(p_1)$$ makes clear, intuitive sense – $$sigma(p)$$ is the effective area of an atom as seen by an electron of momentum $$p$$. However, I’m confused by the detailed-balance equation for this process, for which the relation reads

$$dw(p_2, p_2′; B| p_1; A) d^3p_1 g(p_1) = dw(p_1; A | p_2, p_2′; B) d^3p_2 g(p_2) d^3p’_2 g(p’_2)$$

I’ve seen the three-body rate expressed as

$$Gamma_{eii} = N^2_e n_A int d^3p’_2d^3p_2 f(p_2)f(p’_2) dw(p_1; A | p_2, p_2′; B)\ = N^2_e n_A int d^3p_1 f(p_2)f(p’_2) dw(p_2, p_2′; B | p_1; A ) frac{g(p_1)}{g(p_2)g(p’_2)}$$

The second line of this looks almost nonsensical – the integration in the non-$$p_1$$ variables must somehow be implicily carried out via $$dw$$, which conceals a number of Dirac deltas.

What exactly does one mean by $$dsigma$$ and $$dw$$ – are they differential forms on the manifold of all possible scattering parameters, and in this context, where do the density of states arise from in detailed balance? (i.e. why are they present in the detailed-balance equation, but absent from the total scattering rate integral?)

## Related Questions

### Is the string-net model Hermitian?

2  Asked on February 16, 2021

### Why the fields of the quasi-TEM mode of microstrip line tend to concentrate in the dielectric as frequency goes higher?

0  Asked on February 16, 2021 by george-c

### Significance of Diagonalization in Degenerate perturbation Theory

1  Asked on February 16, 2021

### Dark energy and virtual particles

1  Asked on February 16, 2021 by ilya-grushevskiy

### Moving boundary work

3  Asked on February 16, 2021 by ashish-sona

### Trouble with enterpreting Faraday’s Law

1  Asked on February 16, 2021

### Why quarks are confined? Why can’t they be found in unbound states?

3  Asked on February 16, 2021

### Non-Periodic Boundary conditions for wave equation

0  Asked on February 16, 2021

2  Asked on February 16, 2021 by beastx

### Changing to action-angle variables to reduce the degrees of freedom

0  Asked on February 16, 2021 by loprocto

### Why do two pendulums with the same period meet at this angle?

2  Asked on February 16, 2021 by tmvaz

### Why the direction of propagation of wave is perpendicular to wavefront at each point

1  Asked on February 16, 2021 by shivansh-j

### Why is Earth’s atmosphere made almost entirely of $rm N_2$ and $rm O_2$ molecules rather than any of the nitrogen oxides?

1  Asked on February 16, 2021

### Why does inductor current lag the applied voltage at its terminal by 90 degrees?

5  Asked on February 16, 2021 by vkj

### Circular motion, kinetic friction and tension

1  Asked on February 16, 2021 by diana-arosemena

### Lorentz non-invariance of $3$-acceleration

1  Asked on February 16, 2021 by constantin

### Will the gravity of two non-ionic tungsten atoms at rest a hundred light years apart (many relevant details omitted) eventually draw them together?

1  Asked on February 16, 2021 by owlsupport

### How is the set of states $Q$ logically replaced by a Hilbert space?

1  Asked on February 16, 2021 by jack-rock

### Why an undoped heterojunction causes potential barrier?

1  Asked on February 16, 2021

### Vacuum expectation vs ensemble average

0  Asked on February 16, 2021 by thomas-bakx