SELECTIVE DISORDER PRESERVES THE WHOLE: Sublattice Melting, Cooperative Route Formation, and Liquid-Like Transport Within an Ordered Solid

SELECTIVE DISORDER PRESERVES THE WHOLE:

Sublattice Melting, Cooperative Route Formation, and Liquid-Like Transport Within an Ordered Solid

DOI: [To be assigned]
John Swygert
July 16, 2026

Abstract

Superionic conductors present an apparent contradiction: selected ions move through them with liquid-like freedom while the surrounding material retains a solid crystalline structure. Recent work by Niyogi, Nakamura, Kobayashi, Ando, and Kawasaki uses a chemically neutral minimal model to connect this behavior to sublattice melting, anharmonic lattice fluctuations, and spatially heterogeneous cooperative transport. Their results indicate that mobile carriers do not merely perform independent hops between fixed sites. Near the transition into the superionic state, they move collectively through string-like and locally concentrated routes while the host lattice remains substantially ordered.
This paper interprets those findings through The Simplest Theory of Everything and Other Things, or TSTOEAO. The central proposition is that a system can preserve its larger identity by distributing different equilibrium responsibilities among its components. The host framework carries structural order. The carrier population carries mobility and controlled disorder. The result is not a failure of equilibrium but a division of equilibrium labor in which disorder is selectively located where transport is required.
A domain-specific extension of the TSTOEAO expression V = E \times Y is proposed for superionic systems, along with a conceptual selective-disorder diagnostic and a set of testable predictions. The central conclusion is that optimal transport may occur neither in maximum order nor in complete melting, but within a bounded intermediate regime where one subsystem preserves form while another is permitted to become fluid.

1. Introduction

The conventional opposition between solid and liquid is useful at the macroscopic level but incomplete when applied to internally differentiated systems.
A solid is generally associated with structural order, fixed relative positions, and resistance to deformation. A liquid is associated with mobility, rearrangement, and the loss of persistent positional order. Superionic conductors combine important aspects of both. Their crystalline framework remains intact while selected ions move through the framework with unusually high diffusivity.
The resulting material is not simply half solid and half liquid. It is a structured system in which different populations occupy different dynamical regimes at the same time.
Recent research led by the University of Osaka has clarified the physical basis of this state using a minimal model rather than beginning with the complicated chemistry of a particular material. The researchers represented the system as a rigid host lattice stabilized by strong short-range repulsion and a softer carrier sublattice governed by longer-range interactions. As temperature increased, the carrier population lost its ordered arrangement and became fluid-like before the host lattice melted. This produced an intermediate superionic regime in which mobile carriers flowed through the interstitial spaces of an ordered solid framework [1,2].
Near this transition, carrier transport was spatially heterogeneous, cooperative, and strongly anharmonic. The carriers did not behave merely as independent particles jumping through a static background. Their movement developed through correlated, string-like patterns associated with local softness and collective rearrangement [1].
These findings align closely with the TSTOEAO treatment of gradients, boundary conditions, routed correction, cost-location, and equilibrium targets. More importantly, they reveal a general systems principle:
The whole can preserve its identity by assigning structural order to one subsystem and mobility-producing disorder to another.
Under this interpretation, disorder is not necessarily the opposite of functional equilibrium. Properly bounded and located, it can become one of equilibrium’s operating mechanisms.

2. The Experimental and Computational Foundation

Niyogi and colleagues constructed a chemically neutral binary model containing two distinguishable particle populations.
The first population formed the host lattice. Strong, short-range steric interactions stabilized this framework and preserved its crystalline arrangement.
The second population consisted of smaller carrier particles occupying interstitial regions within the host framework. Softer, longer-range interactions allowed the carriers to form their own ordered sublattice at lower temperatures while also permitting that sublattice to lose order independently of the host.
The simulations identified three broad regimes:
A low-temperature crystalline regime in which both the host and carrier populations remained ordered.
An intermediate sublattice-melt regime in which the carriers became mobile and fluid-like while the host remained crystalline.
A fully molten regime in which both populations lost their ordered structure.
The intermediate regime is the important one. It supports rapid carrier transport without sacrificing the mechanical identity supplied by the host lattice.
The study further found that transport near sublattice melting cannot be described adequately as a collection of isolated, mean-field hopping events. Carrier motion became spatially heterogeneous. Some regions were dynamically active while others remained comparatively frozen. The active carriers frequently moved in cooperative, string-like arrangements.
Increasing anharmonicity—motion that departs from an ideal spring-like response—softened the local carrier environment and promoted cooperative movement. Adjusting particle density shifted the onset and character of sublattice melting. A three-dimensional model of silver iodide reproduced similar qualitative transport regimes, supporting the broader relevance of the mechanism [1,2].
The experimental contribution is therefore more than the observation that ions can move through solids. It identifies a relationship among four features:
selective sublattice melting;
retained host-lattice order;
local anharmonic softness;
cooperative route formation.
Together, these features produce high mobility inside a structure that preserves its larger form.

3. The TSTOEAO Interpretation

TSTOEAO commonly examines physical behavior through the sequence:
gradient → boundary condition → correction → cost-location → equilibrium target
The superionic state follows this sequence with unusual clarity.

Gradient

The carrier population is subjected to thermal, structural, and potentially electrochemical gradients that favor redistribution.

Boundary condition

The host lattice defines the interstitial spaces, bottlenecks, local stiffness, and available directions through which the carriers may move.

Correction

The carriers redistribute through the available route-space. Near sublattice melting, this correction becomes cooperative rather than merely individual.

Cost-location

The loss of positional order is concentrated primarily within the carrier sublattice. The host framework does not pay the full structural cost of melting.

Equilibrium target

The system achieves rapid ionic transport while retaining the mechanical and crystalline framework necessary for continued operation.
This is a form of routed equilibrium. The system does not eliminate motion, thermal agitation, or disorder. It locates them within the subsystem capable of using them productively.
The host lattice maintains continuity.
The carrier sublattice performs redistribution.
The material remains recognizably solid because the subsystem responsible for global structure stays ordered enough to preserve that identity.

4. Order Is Not Identical to Equilibrium

A perfectly ordered state is not automatically the most functional equilibrium state.
At sufficiently low temperature, both the host and carrier sublattices may remain strongly ordered. The material is structurally stable, but carrier mobility is low. The system has order without achieving the transport target.
At sufficiently high temperature, both sublattices may melt. Carrier mobility may remain high, but the host loses the structure that distinguishes the solid electrolyte and makes it mechanically useful.
The functional regime lies between these extremes.
This demonstrates an important distinction:
Equilibrium quality must be evaluated against the target of the system, not against the amount of order present.
For a superionic conductor, the target is not maximum crystallinity. Nor is it maximum mobility without structural constraint. The target is the coexistence of sufficient host stability and sufficient carrier mobility.
The low-temperature crystal may therefore be highly ordered but transport-poor.
The fully molten state may be highly mobile but structure-poor.
The superionic state reaches a more useful balance by allowing the two subsystems to occupy different degrees of order.
Under TSTOEAO, this can be described as an optimal system-equilibrium-quality band rather than a single universal state. The desirable band exists where the host remains above its structural threshold while carrier transport rises above its functional threshold.

5. The Division of Equilibrium Labor

The superionic state demonstrates what may be called a division of equilibrium labor.
In complex systems, every component does not need to perform the same function or occupy the same state. One subsystem may preserve boundaries while another transports material. One may retain memory while another explores alternatives. One may stabilize form while another absorbs fluctuation.
In the present case:
the host lattice carries structural persistence;
the carrier sublattice carries redistribution;
the host defines the route-space;
the carriers explore and use that route-space;
the host limits global disorder;
the carriers convert local disorder into transport.
The entire material remains functional because these responsibilities are separated without becoming disconnected.
This leads to a broader principle:
Stable systems need not suppress all disorder. They may preserve global equilibrium by assigning bounded disorder to the subsystem through which correction must occur.
The disorder is not arbitrary. It is constrained by the host architecture, concentrated within the carrier population, and directed toward a useful transport outcome.
This is why the phrase “selective disorder” is more accurate than simply calling the state partially melted. The loss of order is selective in population, location, function, and extent.

6. A Domain-Specific Expression

The foundational TSTOEAO expression is:
[ V = E \times Y ]
where value or viable realization depends upon encoded equilibrium and yield.
For the present domain, a useful specialization is:
[ V_{SC} = E_H \times Y_R ]
where:
V_{SC} is the functional value of the superionic state;
E_H is the retained encoded equilibrium of the host framework;
Y_R is the yield produced by routed carrier transport.
This is not proposed as a replacement for detailed conductivity equations, diffusion models, or thermodynamic descriptions. It is a system-level representation of the functional dependence.
If the host framework loses its encoded structural equilibrium, then E_H falls and the material loses mechanical identity.
If the carrier sublattice remains locked in place, then Y_R remains low even though the host is strongly ordered.
Functional value rises only when the host preserves sufficient structure and the carriers produce sufficient routed transport.
This product formulation exposes why neither extreme is optimal:
Maximum host order with negligible transport produces low functional yield.
Maximum carrier freedom with a collapsed host produces low retained equilibrium.
Selective sublattice melting can preserve both terms at useful levels.
The highest-value regime is therefore not necessarily the point of maximum order or maximum motion. It is the regime in which retained structure and routed yield multiply most favorably.

7. A Conceptual Selective-Disorder Diagnostic

A simple conceptual diagnostic may help distinguish useful selective disorder from either rigid immobilization or general structural collapse.
Let:
O_H represent normalized host-lattice order;
O_C represent normalized carrier-sublattice order;
M_C represent normalized carrier mobility.
A selective-disorder function may then be written as:
[ F_{SD} = O_H(1-O_C)M_C ]
This is a heuristic systems diagnostic, not an established material law.
Its purpose is to express the conditions required for productive selective disorder:
host order must remain high;
carrier order must decrease;
carrier mobility must actually increase.
If both populations remain ordered, then (1-O_C) is small and the selective-disorder function remains low.
If both populations melt, then O_H collapses and the function again becomes low.
If the carrier sublattice loses order without producing meaningful mobility, then M_C remains low.
The function becomes large only when host order, carrier disorder, and carrier mobility coexist.
The corresponding selective-disorder window can be represented by three threshold conditions:
[ O_H \geq O_{H,\min} ]
[ O_C \leq O_{C,\mathrm{melt}} ]
[ R_C \geq R_{C,\mathrm{conn}} ]
where R_C represents the connectivity of carrier routes.
The first condition preserves the whole.
The second frees the mobile subsystem from excessive positional constraint.
The third ensures that the resulting motion forms usable transport routes rather than disconnected local fluctuations.

8. Cooperative Motion and Route Formation

The observed string-like movement is especially important because it challenges the simplest picture of ionic conduction as independent particles repeatedly hopping between fixed sites.
Individual hopping may still occur, but the simulations indicate that the decisive transition includes cooperative and spatially heterogeneous behavior. One carrier’s movement changes the local conditions encountered by others. A local rearrangement can open, soften, narrow, or redirect neighboring possibilities.
The relevant object is therefore not only the particle or the vacant site. It is the evolving route.
A route can be understood as a temporarily connected sequence of locally permitted transitions. When multiple carrier movements become correlated, the system does not merely contain many independent events. It develops a collective path through its internal boundary structure.
This suggests a route-space interpretation of superionic conduction:
Thermal and structural fluctuations alter local constraints.
Some interstitial regions become softer or more accessible than others.
Carrier movement through one region changes the local environment.
Neighboring carriers respond to the altered route conditions.
Connected strings of movement emerge.
The collective route carries redistribution more efficiently than isolated hopping alone.
The route may be temporary. It need not become a permanent physical channel. Its importance lies in the coordinated sequence of boundary crossings that becomes available during the motion.
This aligns with the TSTOEAO concept that transport often depends not only on the magnitude of a gradient but on the availability, continuity, and cost of routes through which the gradient can be corrected.

9. Anharmonicity as Boundary Softening

In an ideal harmonic description, particles displaced from equilibrium experience restoring forces proportional to the displacement. The local environment behaves approximately like a set of springs.
The reported superionic dynamics become increasingly anharmonic near the relevant transition. The local response is no longer adequately represented by a fixed spring-like landscape. Fluctuations become asymmetric, nonlinear, and structurally heterogeneous.
Within TSTOEAO, this can be interpreted as boundary softening.
The host lattice does not disappear, but the effective restrictions experienced by carriers become more variable. Bottlenecks can widen temporarily. Local energy barriers can fall. Regions that were previously unfavorable can become accessible long enough for cooperative movement to pass through them.
Anharmonicity is therefore not merely noise added to an otherwise ideal lattice. It can participate in forming the route-space.
This does not mean that more anharmonicity is always better. Excessive softening may eventually destabilize the host itself. The useful regime should be bounded:
Enough anharmonicity to soften carrier routes, but not enough to erase the framework that defines and contains those routes.
This again places the functional state near a boundary between excessive rigidity and excessive structural loss.

10. Cost-Location

Every transport process carries costs.
For the carriers to move freely, some positional order must be surrendered. Energy barriers must be crossed or softened. Local environments must fluctuate. Correlated rearrangements must occur.
The important issue is not whether a cost exists but where the system locates it.
A conventional full melting transition distributes structural disorder throughout the material. The host and mobile populations both surrender their ordered arrangements.
Sublattice melting places the principal disorder cost within the carrier subsystem while preserving the host framework.
The system therefore avoids paying for mobility with complete structural collapse.
This is a physical example of a general TSTOEAO rule:
A favorable system does not necessarily eliminate correction cost; it places the cost where it produces the needed correction while doing the least damage to the larger equilibrium.
The carriers pay the positional-order cost.
The host retains structural continuity.
The combined system receives transport without surrendering its entire form.
This is selective cost-location operating as a material property.

11. Testable Predictions

The interpretation developed here produces several testable expectations.

11.1 Route connectivity should predict transport onset better than average local hopping alone

If superionic transport is fundamentally cooperative, the sharp rise in conductivity should correlate strongly with the emergence of system-spanning or repeatedly connected carrier-motion strings.
A material could display numerous local hopping events without achieving high net transport if those events remain disconnected or repeatedly reverse.
The decisive variable should be the connectivity and persistence of the collective route-space.

11.2 The best transport regime should maximize separation between host and carrier order

High-performance superionic behavior should occur where the host retains substantial order while the carrier sublattice exhibits a pronounced loss of positional order.
The useful state should therefore be characterized by a large but bounded separation between the order parameters of the two populations.
Too little separation produces immobilization.
Too much host-order loss produces general melting or structural failure.

11.3 Anharmonicity should have a nonmonotonic optimum

Increasing anharmonicity should initially promote conductivity by softening local constraints and enabling cooperative carrier motion.
Beyond an optimal range, further anharmonicity should begin degrading the host framework, broadening unwanted modes of motion, or destabilizing the material.
Conductivity and mechanical integrity should therefore produce a joint optimum rather than improving indefinitely with lattice softness.

11.4 Local stiffness gradients should steer cooperative transport

If carrier motion follows temporarily softened routes, deliberately patterned stiffness, strain, pressure, composition, or confinement gradients should bias where cooperative strings form.
The route-space should be steerable without requiring a permanent open channel.
This offers a design strategy based on shaping local boundary conditions rather than simply increasing the number of vacancies.

11.5 Defect topology may matter more than defect quantity

Two materials containing the same number of defects may exhibit very different conductivity if one defect arrangement creates connected low-cost routes while the other leaves favorable regions isolated.
The arrangement, orientation, and connectivity of defects should therefore be at least as important as their total concentration.

11.6 Density should control more than available space

The published model already demonstrates that density shifts the onset of sublattice melting [1].
The TSTOEAO interpretation adds that density changes the route grammar of the material. It alters bottleneck dimensions, local stiffness, interaction distances, and the degree to which one carrier’s movement affects neighboring carriers.
Density should therefore control not only whether movement is possible but whether that movement remains isolated or becomes cooperative.

11.7 Conductivity may display route-history effects

Where cooperative routes require correlated rearrangement, the response to a field or temperature change may depend partly on the recent dynamical history of the material.
A previously established route population may decay over a finite period rather than disappearing instantaneously.
This predicts potentially measurable hysteresis, pulse-duration dependence, or differing responses to continuous and intermittent stimulation, even when average temperature and field magnitude are similar.
This prediction is more speculative than the preceding ones and should be tested directly rather than assumed.

11.8 Failure should begin when disorder escapes its assigned subsystem

The functional state depends upon keeping mobility-producing disorder primarily within the carrier sublattice.
Structural degradation should begin when the fluctuations supporting carrier movement increasingly recruit, deform, or disorder the host lattice.
The failure boundary can therefore be defined as the point at which useful selective disorder becomes spreading disorder.
Monitoring cross-correlation between carrier mobility and host displacement may provide an early indicator of that transition.

12. Implications for Materials Design

The conventional design question for a solid electrolyte is often framed as:
How can ions be made to move more easily through a solid?
The present framework suggests a more precise question:
How can one subsystem be brought near its mobility-producing disorder transition while the surrounding framework remains above its structural-stability threshold?
That shifts attention toward the controlled separation of internal regimes.
Potential design variables include:
differences between host and carrier interaction strengths;
carrier density and partial occupancy;
interstitial geometry;
local bottleneck dimensions;
host-lattice stiffness;
controlled anharmonic modes;
defect connectivity;
strain patterns;
pressure;
compositional gradients;
interfaces that seed or guide cooperative routes.
The goal is not to make the entire material softer.
The goal is to soften the correct routes for the correct population while preserving the boundaries that keep the material intact.
This can be stated as a materials-design rule:
Engineer the host for retained identity and the carrier environment for bounded cooperative release.
The strongest material may not be the most rigid one. The fastest conductor may not be the least constrained one. The preferred system is the one that most effectively separates structural responsibility from transport responsibility while maintaining communication between them.

13. Broader Significance

The significance of sublattice melting extends beyond solid-state batteries.
Many functional systems depend upon the coexistence of a stable architecture and a mobile internal population. Biological membranes preserve cellular boundaries while permitting selective transport. Porous materials maintain structural form while allowing molecules to enter, move, react, or exit. Neural systems preserve broad organization while local activity patterns continually reorganize. Social and computational systems maintain persistent rules while allowing agents, information, and decisions to move through them.
These systems should not be treated as physically identical. Their mechanisms, scales, and governing laws differ substantially.
They may nevertheless share a systems-level principle:
Persistence and change do not always need to compromise at every location. They can be divided among interacting subsystems.
The superionic conductor provides a particularly clean physical example because the division is measurable. One sublattice remains ordered. Another loses order. The material’s functionality emerges from their coexistence.
This supports a broader TSTOEAO proposition:
The preservation of a whole may require that selected parts be permitted to move farther from order than the whole itself could survive.
The larger equilibrium is protected not by forcing every component into the same condition, but by correctly allocating stability, motion, disorder, and correction.

14. What This Paper Does Not Claim

The results of Niyogi and colleagues do not prove TSTOEAO as a fundamental physical theory.
Their work was developed independently through molecular-dynamics modeling, analysis of sublattice melting, and comparison with a three-dimensional silver-iodide system. The terminology of routed equilibrium, cost-location, boundary softening, and division of equilibrium labor is the interpretive contribution of this paper.
This paper also does not claim that all ionic conductors operate through an identical mechanism or that all cooperative motion must be string-like. Material-specific chemistry, defects, dimensionality, electrostatic interactions, interfaces, and operating conditions remain important.
The appropriate claim is narrower and more useful:
The reported results provide a strong physical example of a system achieving superior function through selective disorder, cooperative route formation, and the separation of structural and transport responsibilities.
The value of the alignment lies in the predictions and design questions it generates, not in retroactively relabeling the researchers’ work.

15. Conclusion

Superionic conduction reveals that a material can remain solid without requiring all of its internal components to remain solid-like.
The host lattice preserves the structure.
The carrier sublattice relinquishes positional order.
Anharmonic fluctuations soften local boundaries.
Cooperative motion connects those softened regions into transport routes.
The resulting system achieves liquid-like ionic movement without paying the cost of complete structural melting.
Through TSTOEAO, this can be understood as a division of equilibrium labor and an example of selective cost-location. The system preserves its global identity by concentrating mobility-producing disorder within the subsystem responsible for correction.
The central law may be stated simply:
Selective disorder preserves the whole when disorder is bounded, routed, and assigned to the part through which change must occur.
The most functional state is not always the most ordered state.
It is the state in which order and disorder are each placed where they perform the greatest service to the larger equilibrium.

References

Niyogi, Sucharita, Takenobu Nakamura, Genki Kobayashi, Yasunobu Ando, and Takeshi Kawasaki. “Probing Anharmonic and Heterogeneous Carrier Dynamics Across Sublattice Melting in a Minimal Model Superionic Conductor.” Proceedings of the National Academy of Sciences of the United States of America, vol. 123, no. 28, e2605867123, July 7, 2026. DOI: 10.1073/pnas.2605867123.
University of Osaka. “How Ions Flow Like a Liquid Through a Solid Crystal.” Phys.org, July 16, 2026.

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