10 We have also investigated the case β ≪ 1 with all other parameters \(\cal O(1)\) to verify that this case does indeed approach the racemic state at large times (that is, θ, ϕ, ζ → 0 as t → ∞). However, once again the difference in timescales can be observed, with the concentrations reaching equilibration on a faster timescale than the chiralities, due to the different magnitudes

of eigenvalues (Eq. 4.28). New Simplifications of the System We return to the Eqs. 2.35–2.39 in the case δ = 0, now writing x 2 = x and y = y 2 to obtain $$ \frac\rm d c\rm d t = – 2 \mu c + \mu\nu (x+y) – \alpha c(N_x+N_y) , $$ (5.1) $$ \frac\rm d x\rm d t = \mu c – \mu\nu x – \alpha x c + \beta (N_x-x + x_4) – \xi x^2 – \xi x N_x , $$ (5.2) $$ \frac\rm d y\rm d t = \mu c – \mu\nu y – \alpha y c + \beta (N_y-y + y_4) – \xi y^2

selleck products – \xi y N_y , $$ (5.3) $$ \frac\rm d N_x\rm d t = \mu c – \mu\nu x + \beta (N_x-x) – \xi x N_x , $$ (5.4) $$ \frac\rm d N_y\rm d t click here = \mu c – \mu\nu y + \beta (N_y-y) – \xi y N_y , $$ (5.5)which are not closed, since x 4, y 4 appear on the rhs’s of Eqs. 5.2 and 5.3, hence we need to find formulae to determine x 4 and y 4 in terms of x, y, N x , N y . One way of achieving this is to expand the system to include other properties of the distribution of cluster sizes. For example, equations governing the mass of crystals in each chirality can be derived as $$ \frac\rm d \varrho_x\rm d t=2\mu c-2\mu\nu x+2\alpha c N_x , \quad \frac\rm d \varrho_y\rm d t=2\mu c-2\mu\nu y+2\alpha c N_y . $$ (5.6)These introduce no more new new quantities into the macroscopic system of equations, and do not rely on knowing x 4 or y 4, (although they do require knowledge of x and y). In the remainder of this section we consider various potential formulae for x 4, y 4 in terms of macroscopic quantities so that a macroscopic system can be constructed. We then analyse such macroscopic systems in two specific limits to show that predictions

relating to symmetry-breaking can be made. Reductions during The equations governing the larger cluster sizes x k , y k , are $$ \frac\rm d x_2k\rm d t = \beta( x_2k+2 – x_2k ) – (x_2k-x_2k-2)(\alpha c + \xi x) ; $$ (5.7)in general this has solutions of the form \(x_2k = \sum_j A_j(t) \Lambda_j^k-1\), where Λ j are parameters (typically taking values between unity (corresponding to a steady-state in which mass is being added to the distribution) and \(\frac\alpha c+\xi x\beta\) (the equilibrium value); and A j (t) are time-dependent; for some Λ j , A j will be constant. We assume that the distribution of each chirality of cluster is given by $$ x_2k = x \left( 1 – \frac1\lambda_x \right)^k-1 ,\qquad\qquad y_2k = y \left( 1 – \frac1\lambda_y \right)^k-1 , $$ (5.8)since solutions of this form may be steady-states of the governing Eq. 5.7.