Effective potential and chiral symmetry breaking

The nonequilibrium effective potential is calculated for the Frank model of spontaneous mirror-symmetry breaking in chemistry in which external noise is introduced to account for random environmental effects. The well-mixed limit, corresponding to negligible diffusion, and the case of diffusion in two space dimensions are studied in detail. White noise has a disordering effect in the former case, whereas in the latter case a phase transition occurs for external noise exceeding a critical intensity which racemizes the system.

Figure 1

The Frank model potential $V\left(\eta \right)/b$, Eq. (12), displaying the racemizing tendency of the direct chiral monomer production $A\to L,A\to D$. The sequence of curves from bottom to top corresponds to $r=0,0.2,0.4$ and $r=0.5$. For $r>0.5$, the racemic state is the unique stable final outcome.

Figure 2

Tree-level potential: the enantiomeric excess $\eta$ as a function of $r=a/b$. We display the positive branch $\eta =+\sqrt{1-2r}$. There is also a mirror-image negative branch which completes the figure and gives the classic bifurcation diagram. Note $\eta =0$ for all $r>\frac{1}{2}$.

Figure 3

Tree potential subject to small chiral bias, Eq. (17). The dashed curve corresponds to zero bias. For ${ϵ}^{\prime }>0$, the point $\eta =1$ is the absolute minimum of the potential. For the sequence of solid curves below dashed curve, from top to bottom, ${ϵ}^{\prime }=0.1$, 0.2, and 0.5. For ${ϵ}^{\prime }<0$, the curves will be tilted in the opposite sense.

Figure 4

The one-loop effective potential $V_{\mathcal{A}}/b$ for the Frank model, Eq. (27). The final enantiomeric excess $\eta$ corresponds to the absolute minima of the potential and decreases in absolute value below unity as the noise strength increases. The curves from bottom to top correspond to $\mathcal{A}/2b=0.0$, 0.05, 0.1, 0.2, and 0.3.

Figure 5

The biased one-loop effective potential $V_{\mathcal{A}}/b$ for the Frank model with ${ϵ}^{\prime }=0.1$. The sequence of curves from bottom to top corresponds to $\mathcal{A}/2b=0.0$, 0.05,0.1, and 0.2. Compare and contrast to Fig. 4.

Figure 6

(Case I) With ${\mu }_{diff}={\mu }_0$. The $d=2$ one-loop effective potential $V_{\mathcal{A}}/b$ for the Frank model. The $\mu$ dependence drops out in this case. The range of the noise parameter $\frac{\mathcal{A}}{16\pi D}=\left(0,0.1,0.3,0.5,0.7\right)$ corresponds to the sequence of curves from bottom to top.

Figure 7

(Case II) The $d=2$ one-loop effective potential $V_{\mathcal{A}}/b$. With ${\mu }_{diff}\le \mu <{\mu }_0$, specifically: $l_{diff}=10\times l_0$, and $l=5\times l_0$. Noise parameter $\frac{\mathcal{A}}{16\pi D}=\left(0,0.01,0.02,0.03,0.04\right)$; sequence of curves from bottom to top.

Figure 8

(Case III) The $d=2$ one-loop effective potential $V_{\mathcal{A}}/b$. With $\mu <{\mu }_{diff}<{\mu }_0$. Specifically, $l_{diff}=5\times l_0$ and $l=10\times l_0$. Noise parameter $\frac{\mathcal{A}}{16\pi D}=\left(0,0.01,0.02,0.03,0.04\right)$; sequence of curves from bottom to top.

Figure 9

(Case IV) The $d=2$ one-loop effective potential $V_{\mathcal{A}}/b$. With $\mu ={\mu }_{diff}$ and $l_{diff}=1000\times l_0$. Noise parameter $\frac{\mathcal{A}}{16\pi D}=\left(0,0.005,0.01,0.015\right)$; sequence of curves from bottom to top.

Figure 10

The biased $d=2$ one-loop effective potential $V_{\mathcal{A}}/b$ for ${ϵ}^{\prime }=0.1$ and $\mu ={\mu }_{diff}$ and $l_{diff}=1000\times l_0$. Noise parameter $\frac{\mathcal{A}}{16\pi D}=\left(0.005,0.01,0.015,0.02\right)$; sequence of curves from bottom to top.