standard Pechukas-Yukawa model, as detailed in [14], we arrive at the following generalised system of equations; ˙ x$_{n}$ = ∂x$_{n}$ ∂λ = v$_{n}$ + ˙ δh$_{nn}$, ˙ v$_{n}$ = ∂v$_{n}$ ∂λ = ∑ k = n [ 2 | l$_{nk}$ | 2 ( x$_{n}$ − x$_{k}$ ) 3 + l$_{nk}$ ˙ δh$_{kn}$ − ˙ δh$_{nk}$l$_{kn}$ ( x$_{n}$ − x$_{k}$ ) 2 ] , (3) ˙ l$_{nm}$ = ∂l$_{nm}$ ∂λ = ∑ k = m,n [ l$_{nk}$l$_{km}$$^{(}$ 1 ( x$_{n}$ − x$_{k}$ ) 2 − 1 ( x$_{m}$ − x$_{k}$ ) 2 ) + ( x$_{n}$ − x$_{m}$ )( l$_{nk}$ ˙ δh$_{km}$ − l$_{km}$ ˙ δh$_{nk}$ ) ( x$_{m}$ − x$_{k}$ )( x$_{n}$ − x$_{k}$ ) ] + ˙ δh$_{nm}$ ( v$_{m}$ − v$_{n}$ ) + l$_{nm}$ ( ˙ δh$_{nn}$ − ˙ δh$_{mm}$ ) ( x$_{n}$ − x$_{m}$ ) . where v$_{n}$ ( λ ) = 〈 n | ZH$_{b}$ | n 〉 and l$_{nm}$ = ( x$_{n}$ − x$_{m}$ ) 〈 n | ZH$_{b}$ | m 〉 for n = m . These generalised equations can be used to describe a wider array of physical systems because of the inclusion of the δh ( λ ) term without any additional assumptions or approximations. Note that if the noise term δh ( λ ( t )) is identically zero the system of equations (3) simply reduces to the standard Pechukas-Yukawa equations. The equations (3) describe the dynamics of the energy eigenvalues of the Hamiltonian (2), but also correspond to the classical dynamics of a 1D interacting gas, where the n th particle has position x$_{n}$ ( λ ) and velocity v$_{n}$ ( λ ) and the strength of the inter-particle force between the n th and the m th particles is described by l$_{nm}$ ( λ ). In order to close the system of equations (3), we need to consider the nature of the noise term δh ( λ ). In general, noise in any physical system arises from a number of independent sources and therefore as a consequence of the central-limit theorem it seems reasonable to assume that the sum of their effects will result in a random Hamiltonian with independent Gaussian distributed elements. Such a Hamiltonian will be drawn from one of the Gaussian ensembles of random matrix theory. Recent measurements of the low frequency flux noise in superconducting flux qubits exhibit a coloured noise spectrum [15]. We therefore assume that the noise term, δh ( λ ), evolves in time as a Ornstein-Uhlenbeck type process, which is a simple example of a random process with a coloured spectrum: where τ is a correlation time, ϵ is the noise amplitude, and η ( λ ) is a random matrix valued stochastic process where 〈 η ( λ ) 〉 = 0 and 〈 η ( λ ) η ( λ $^{′}$) 〉 = δ ( λ − λ $^{′}$). ˙ δh ( λ ) = − τ δh ( λ ) + ϵη ( λ ) (4) 4