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Equation 3 · Part 2 · Decoherence: The Quiet Selection That Makes the World Look Solid

Symbol x

ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]
xx

What this part means

x is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

x is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

Joos and Zeh made this quantitative in 1985 by deriving a genuine, non-phenomenological master equation for the reduced density matrix of an object’s centre-of-mass position under repeated recoil-free scattering [ 7 ] . For the off-diagonal elements of that density matrix — the very quantity whose survival or destruction is the entire question of macroscopic superposition — their result, in the short-time, many-collisions regime, takes the form of a simple exponential decay in the separation between the two positions being superposed: ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]. Here Λ\Lambda is what Joos and Zeh call the localization rate: a single number, with units of inverse length squared per unit time, built…

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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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