^{1}and Richard M. Stratt

^{1,a)}

### Abstract

How useful it is to think about the potential energy landscape of a complex many-body system depends in large measure on how direct the connection is to the system’s dynamics. In this paper we show that, within what we call the potential-energy-landscape ensemble, it is possible to make direct connections between the geometry of the landscape and the long-time dynamical behaviors of systems such as supercooled liquids. We show, in particular, that the onset of slow dynamics in such systems is governed directly by the lengths of their *geodesics*—the shortest paths through their landscapes within the special ensemble. The more convoluted and labyrinthine these geodesics are, the slower that dynamics is. Geodesics in the landscape ensemble have sufficiently well-defined characteristics that it is straightforward to search for them numerically, a point we illustrate by computing the geodesic lengths for an ordinary atomic liquid and a binary glass-forming atomic mixture. We find that the temperature dependence of the diffusion constants of these systems, including the precipitous drop as the glass-forming system approaches its empirical mode-coupling transition, is predicted quantitatively by the growth of the geodesic path lengths.

We thank Kenneth Schweizer, David Wales, and Guohua Tao for helpful discussions. This work was supported by NSF Grant Nos. CHE-0518169 and CHE-0131114.

I. INTRODUCTION

II. GEODESIC PATHS IN THE POTENTIAL-ENERGY-LANDSCAPE ENSEMBLE

A. The nature of the optimum paths

B. Some literature connections

C. An application: Geodesic path lengths and diffusion constants

III. FINDING GEODESIC PATHS IN LIQUIDS

IV. MODELS AND METHODS

V. NUMERICAL RESULTS FOR GEODESIC PATHS IN LIQUIDS

VI. CONCLUDING REMARKS

### Key Topics

- Diffusion
- 32.0
- Potential energy surfaces
- 12.0
- Molecular dynamics
- 9.0
- Kinematics
- 6.0
- Green's function methods
- 5.0

## Figures

The potential-energy-landscape ensemble. Both panels show the potential energy as a function of the configuration of the system with the white regions (those lying below the landscape energy ) the allowed regions and the shaded regions the forbidden regions. (Note the different convention from Fig. 1 in Ref. 52.) The lower panel redraws the upper panel as a contour plot, emphasizing the complex, multiply connected, many-dimensional character of the allowed regions.

The potential-energy-landscape ensemble. Both panels show the potential energy as a function of the configuration of the system with the white regions (those lying below the landscape energy ) the allowed regions and the shaded regions the forbidden regions. (Note the different convention from Fig. 1 in Ref. 52.) The lower panel redraws the upper panel as a contour plot, emphasizing the complex, multiply connected, many-dimensional character of the allowed regions.

Geodesics in the potential-energy-landscape ensemble. The upper panel shows a possible shortest path between initial and final configurations and . The path consists of straight-line segments combined with segments skirting the boundary of the forbidden regions, as required by the Kuhn-Tucker theorem. The lower panel shows (as dashed lines) some alternative possibilities for path segments that also obey the Kuhn-Tucker theorem.

Geodesics in the potential-energy-landscape ensemble. The upper panel shows a possible shortest path between initial and final configurations and . The path consists of straight-line segments combined with segments skirting the boundary of the forbidden regions, as required by the Kuhn-Tucker theorem. The lower panel shows (as dashed lines) some alternative possibilities for path segments that also obey the Kuhn-Tucker theorem.

Our algorithm for finding Kuhn-Tucker candidates for geodesic paths. Panel (a) shows how one starts at the initial configuration and heads directly for the final configuration (dark solid line), a procedure that eventually leads to a slight incursion (dotted lines) into a forbidden region, shown here as a circle. This misstep is remedied by following the gradient of the potential until one returns to the boundary of the forbidden region (the black dot). The whole process is then repeated, as shown in panel (b). The final path found by successive applications of the process is shown in (c).

Our algorithm for finding Kuhn-Tucker candidates for geodesic paths. Panel (a) shows how one starts at the initial configuration and heads directly for the final configuration (dark solid line), a procedure that eventually leads to a slight incursion (dotted lines) into a forbidden region, shown here as a circle. This misstep is remedied by following the gradient of the potential until one returns to the boundary of the forbidden region (the black dot). The whole process is then repeated, as shown in panel (b). The final path found by successive applications of the process is shown in (c).

(Color) Different pathways through the two-dimensional Müller-Brown potential energy landscape. The top panel (a) contrasts an energy-landscape geodesic path (for landscape energy ) with the reaction path. The geodesic path goes between two arbitrarily chosen points and and stays reasonably close to the constant-potential-energy contour. The reaction path travels from one minimum (the “reactants” ) to another (the “products” ) by way of yet another minimum [located at] and two different saddle points, and (0.22, 0.30). The bottom panel (b) compares the reaction path with a number of different energy landscape paths between and . The open stars denote the geodesic candidate predicted by our (unoptimized) Kuhn-Tucker path-finding algorithm. The path starts at and heads directly for , turning left when it encounters a barrier, and then resumes its direct path to as soon as it clears the barrier. The filled stars show a shorter path-length geodesic candidate produced by an intermediate level of optimization, and the open circles show the final geodesic path—which differs noticeably from the reaction path, especially in the product region, largely because of the way that the reaction path has to pass through the left saddle point.

(Color) Different pathways through the two-dimensional Müller-Brown potential energy landscape. The top panel (a) contrasts an energy-landscape geodesic path (for landscape energy ) with the reaction path. The geodesic path goes between two arbitrarily chosen points and and stays reasonably close to the constant-potential-energy contour. The reaction path travels from one minimum (the “reactants” ) to another (the “products” ) by way of yet another minimum [located at] and two different saddle points, and (0.22, 0.30). The bottom panel (b) compares the reaction path with a number of different energy landscape paths between and . The open stars denote the geodesic candidate predicted by our (unoptimized) Kuhn-Tucker path-finding algorithm. The path starts at and heads directly for , turning left when it encounters a barrier, and then resumes its direct path to as soon as it clears the barrier. The filled stars show a shorter path-length geodesic candidate produced by an intermediate level of optimization, and the open circles show the final geodesic path—which differs noticeably from the reaction path, especially in the product region, largely because of the way that the reaction path has to pass through the left saddle point.

Energy/temperature relations in the landscape and canonical ensembles for the two different model liquids studied in this paper. Both panels show the configurational temperature from Eq. (1.2) plotted as a function of landscape energy (solid lines with barely visible vertical error bars) and compare that to the canonical average potential energy plotted vs canonical temperature (squares with horizontal error bars). The hatched region in the upper panel is the solid/liquid coexistence region and the dashed gray line in the lower panel is a literature fit to canonical ensemble data for the Kob-Andersen liquid (Ref. 87). Given the scale of the figure, the only place where the reader can probably see an error bar for the landscape results is within the coexistence region of the upper panel. Both panels are for systems with atoms.

Energy/temperature relations in the landscape and canonical ensembles for the two different model liquids studied in this paper. Both panels show the configurational temperature from Eq. (1.2) plotted as a function of landscape energy (solid lines with barely visible vertical error bars) and compare that to the canonical average potential energy plotted vs canonical temperature (squares with horizontal error bars). The hatched region in the upper panel is the solid/liquid coexistence region and the dashed gray line in the lower panel is a literature fit to canonical ensemble data for the Kob-Andersen liquid (Ref. 87). Given the scale of the figure, the only place where the reader can probably see an error bar for the landscape results is within the coexistence region of the upper panel. Both panels are for systems with atoms.

Geodesic path lengths as a function of configurational temperature for the two model liquids studied in this paper. This figure shows how the ratio of , the Euclidean path length (the direct end-to-end distance) to , the geodesic path length, converges as the paths become longer. Both panels are for systems with atoms.

Geodesic path lengths as a function of configurational temperature for the two model liquids studied in this paper. This figure shows how the ratio of , the Euclidean path length (the direct end-to-end distance) to , the geodesic path length, converges as the paths become longer. Both panels are for systems with atoms.

Geodesic path lengths as a function of configurational temperature for the two model liquids studied in this paper. As with Fig. 6, this figure looks at how the square of the ratio of Euclidean to geodesic path length, , decreases with decreasing temperature, but here the emphasis is on the invariance of the results to , the number of atoms, for a given ratio.

Geodesic path lengths as a function of configurational temperature for the two model liquids studied in this paper. As with Fig. 6, this figure looks at how the square of the ratio of Euclidean to geodesic path length, , decreases with decreasing temperature, but here the emphasis is on the invariance of the results to , the number of atoms, for a given ratio.

(Color) Reduced diffusion constants for our two liquids as a function of temperature . Molecular-dynamics derived diffusion constants (points) are compared with landscape geodesic predictions (lines) for the single-component atomic system, (a), and for both the and particles in the Kob-Andersen binary case, (b) and (c), (with Fig. 5 used to translate into ). The location of the literature empirical mode-coupling transition (Refs. 86 and 88) is indicated by the arrow in (c).

(Color) Reduced diffusion constants for our two liquids as a function of temperature . Molecular-dynamics derived diffusion constants (points) are compared with landscape geodesic predictions (lines) for the single-component atomic system, (a), and for both the and particles in the Kob-Andersen binary case, (b) and (c), (with Fig. 5 used to translate into ). The location of the literature empirical mode-coupling transition (Refs. 86 and 88) is indicated by the arrow in (c).

The variation of potential energy along typical geodesic paths in the Kob-Andersen liquid. Shown here are results for nine different geodesics obtained from the nine different configurational temperatures shown in the key. The distance traveled along each path is denoted by s. The sudden jumps and dips at the end points are related to the fact that the potential energies of the beginning and ending configurations are often different from each other and from the landscape energy.

The variation of potential energy along typical geodesic paths in the Kob-Andersen liquid. Shown here are results for nine different geodesics obtained from the nine different configurational temperatures shown in the key. The distance traveled along each path is denoted by s. The sudden jumps and dips at the end points are related to the fact that the potential energies of the beginning and ending configurations are often different from each other and from the landscape energy.

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