{ "id": "epv9h5bbpm", "slug": "cluster-expansion-statistical-mechanics", "title": "Cluster expansion (statistical mechanics)", "updated": "2026-10-25", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation" }, { "id": "physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter" }, { "id": "physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter.cm-structures" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Non-geographic · 1946 to 2000: Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures" } ] } ], "excerpt": "A cluster expansion expresses a lattice configuration's energy as a sum over cluster correlation functions, letting alloy thermodynamics be sampled after training on few structures.", "snippet": "A cluster expansion expresses a lattice configuration's energy as a sum over cluster correlation functions, letting alloy thermodynamics be sampled after training on few structures.", "node": "physical.physics.matter-and-radiation.condensed-matter.cm-structures", "markdown": "# Cluster expansion (statistical mechanics)\n\nA cluster expansion expresses an energy or other scalar property of a crystalline lattice configuration as a sum over cluster correlation functions, so that thermodynamics of an alloy can be sampled after training on only a small subset of configurations. It is a generalized [Ising model](https://www.edgepedia.org/ising-model): effective cluster interactions (ECIs) multiply orthogonal basis functions defined on clusters of lattice sites, with symmetrically equivalent clusters grouped into orbits that share one ECI.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) The method sits between density-functional theory (DFT), which supplies energies of tens to thousands of ordered structures, and statistical thermodynamics, for which the fitted expansion serves as a Hamiltonian for Monte Carlo simulation of phase behavior.[2](https://par.nsf.gov/servlets/purl/10324364) The motivation is combinatorial: a binary system of N atoms has roughly \\( 2^{N} \\) configurations, so a 100-atom cell would require examining on the order of \\( 10^{30} \\) configurations.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001)\n\n| Key fact | Value |\n|---|---|\n| Model form | Generalized Ising model; energy as ECIs times orthogonal cluster basis functions[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) |\n| Typical converged alloy CE | About 10 to 20 ECIs from 30 to 50 ordered structures[3](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html) |\n| Fitting problem | Linear system \\( y = X \\cdot w \\) (targets, ECIs, design matrix)[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) |\n| Achievable cross-validation error | About 1 meV/atom (MSCE, Mg-Zn) to about 15 meV/atom (CrCoNi pair/triplet fits)[4](https://www.nature.com/articles/s41524-023-01029-0) • [5](https://www.nature.com/articles/s41524-024-01338-y) |\n| Practical failure threshold | CE tends to fail when the number of ECIs exceeds 80[6](https://bsg.byu.edu/docs/papers/PhysRevB-96-014107.pdf) |\n| Software | ATAT, UNCLE, CLEASE, CASM, ICET, SMOL[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) |\n\n## How it works\n\nThe defining equation of a cluster expansion on a lattice is \\( E(\\sigma) = E_{\\mathrm{CE}}(\\sigma) = \\sum_{f} J_{f} \\, \\Pi_{f}(\\sigma) \\), a sum over cluster figures f (pairs, triplets, quadruplets, and so on) with ECIs \\( J_{f} \\) multiplying figure functions \\( \\Pi_{f}(\\sigma) \\). The expansion is exact when all figures are included; published proofs establish that all configurational energies can be mapped if the model includes all possible cluster types.[7](https://indico.ictp.it/event/a12198/session/59/contribution/38/material/0/0.pdf) In the icet notation, a property is written \\( Q = \\sum_{\\alpha} m_{\\alpha} J_{\\alpha} \\langle \\Gamma_{\\alpha'}(\\boldsymbol{\\sigma}) \\rangle_{\\alpha} \\), where \\( m_{\\alpha} \\) is the multiplicity of orbit \\( \\alpha \\).[8](https://icet.materialsmodeling.org/dev/get_started/cluster_expansions.html)\n\nThe mathematical core is the 1984 generalized cluster description of Sanchez, Ducastelle, and Gratias, which builds an orthogonal basis in the multidimensional space of discrete spin (occupation) variables; the expectation values of the basis functions are the multisite correlation functions, which form an independent set of variational parameters for the free energy.[9](https://doi.org/10.1016/0378-4371%2884%2990096-7) In complete-basis expansions the ECIs are projections of the energy onto the basis, and the configurational energy is a [homogeneous function](https://www.edgepedia.org/homogeneous-function) of degree one in the correlation functions, with the interactions being the Euler derivatives of the energy with respect to them; this distinguishes true cluster expansions from phenomenological generalized-Ising fits with constant interactions.[10](https://link.springer.com/content/pdf/10.1007/s11669-017-0521-3.pdf) Sanchez later showed that the 1984 cluster basis is a multidimensional discrete [Fourier transform](https://www.edgepedia.org/fourier-transform), while his 1993 variable-basis formalism corresponds to a multidimensional discrete wavelet transform.[11](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.224202) A recent refinement shows that any orthonormal-basis cluster expansion admits a unique, basis-independent decomposition, the cluster decomposition, identifiable with a functional ANOVA (Sobol) decomposition, which matters because for three or more components the numerical values of CE coefficients depend non-trivially on the basis chosen.[5](https://www.nature.com/articles/s41524-024-01338-y)\n\n## How it is done\n\nConstruction proceeds in four steps: choose the lattice and cluster pool, generate training structures, fit the ECIs, and validate.\n\n**Cluster selection.** Cutoff radii per cluster order determine how many ECIs enter the model. Too-small cutoffs cause underfitting and too-large cutoffs overfitting; a good starting cutoff is on the order of the lattice parameter, and cutoffs larger than three lattice parameters are very rarely needed.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001)\n\n**Training set.** Structures are typically ordered supercells whose formation enthalpies, \\( \\Delta H_{f}(\\sigma) = E_{\\mathrm{tot}}(\\sigma) - x E_{\\mathrm{tot}}(\\text{pure A}) - (1-x) E_{\\mathrm{tot}}(\\text{pure B}) \\), rather than raw total energies, are the fitting targets.[7](https://indico.ictp.it/event/a12198/session/59/contribution/38/material/0/0.pdf) Cross-validation, leaving out data points, selects the best cluster combination, and the trace of the ECI covariance matrix identifies configurations that maximize uncertainty reduction; both ideas are implemented in ATAT.[2](https://par.nsf.gov/servlets/purl/10324364)\n\n**Fitting.** Training is the linear problem \\( y = X \\cdot w \\), where y holds target observables, w the unknown ECIs, and X the design matrix.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) Exact inversion is not a reliable strategy; exactly-inverted solutions typically predict poorly.[7](https://indico.ictp.it/event/a12198/session/59/contribution/38/material/0/0.pdf)\n\n**Sampling.** The fitted expansion becomes a Hamiltonian for Monte Carlo simulation, producing free energies and short-range-order parameters as functions of temperature and concentration.[3](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html) In ATAT, construction is performed by the MAPS code and [Monte Carlo](https://www.edgepedia.org/monte-carlo) by the EMC2 code.[3](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html) The choice of ensemble matters: the semi-grand-canonical (SGC) ensemble cannot sample across miscibility gaps because one chemical potential maps to two concentrations, while the variance-constrained semi-grand-canonical (VCSGC) ensemble can.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001)\n\nFor a conventional binary alloy, a well-converged expansion uses about 10 to 20 ECIs fitted to roughly 30 to 50 ordered structures.[3](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html) High-entropy alloys need far more: converged CEs for NbTiVZr, HfNbTaTiZr, and AlHfNbTaTiZr required 2,984, 1,970, and 4,000 DFT structures respectively.[12](https://arxiv.org/html/2403.18298) Reported cross-validation errors span about 1 meV/atom for MSCE fits of Mg-Zn,[4](https://www.nature.com/articles/s41524-023-01029-0) through 12.8 meV/atom (triplet) and 14.9 meV/atom (pair) for [CrCoNi](https://www.edgepedia.org/crconi) fits using 500 training structures,[5](https://www.nature.com/articles/s41524-024-01338-y) to below 10 meV/atom for GNN-accelerated high-throughput fits.[12](https://arxiv.org/html/2403.18298) On the failure side, CE models tend to fail when the number of ECIs exceeds 80.[6](https://bsg.byu.edu/docs/papers/PhysRevB-96-014107.pdf)\n\n## Origin\n\nThe lineage begins with the cluster variation method (CVM), formulated by Ryoichi Kikuchi in \"A Theory of Cooperative Phenomena\" ([Physical Review](https://www.edgepedia.org/physical-review), 1951), which generalized the Bethe model to describe order–disorder phenomena in three-dimensional lattices.[13](https://doi.org/10.1103/physrev.81.988) • [2](https://par.nsf.gov/servlets/purl/10324364) In CVM the variational parameters are probability distributions over clusters; the restructuring of CVM into the modern cluster expansion, in which cluster correlation functions are the variational parameters, came out of the Sanchez group's work.[2](https://par.nsf.gov/servlets/purl/10324364)\n\nThe formalization is the 1984 paper \"Generalized cluster description of multicomponent systems\" by J.M. Sanchez, F. Ducastelle, and D. Gratias in Physica A.[9](https://doi.org/10.1016/0378-4371%2884%2990096-7) Sanchez's 1993 paper \"Cluster expansions and the configurational energy of alloys\" ([Physical Review B](https://www.edgepedia.org/physical-review-b)) introduced the variable-basis formalism.[14](https://doi.org/10.1103/physrevb.48.14013) Fitting ECIs to first-principles energies of a small set of ordered compounds is known as the structure inversion method.[3](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html) Later methodological contributions include the Bayesian cluster expansion of Tim Mueller and [Gerbrand Ceder](https://www.edgepedia.org/gerbrand-ceder) (Physical Review B, 2009),[15](https://doi.org/10.1103/physrevb.80.024103) Bayesian compressive sensing by Lance J. Nelson and colleagues (Physical Review B, 2013),[16](https://doi.org/10.1103/physrevb.88.155105) the mixed-basis expansion with long-range strain terms by [Alex Zunger](https://www.edgepedia.org/alex-zunger), L G Wang, Gus L W Hart, and Mahdi Sanati (2002),[17](https://doi.org/10.1088/0965-0393/10/6/306) the optimal truncation procedure of Nikolai A. Zarkevich and D. D. Johnson (Physical Review Letters, 2004),[18](https://doi.org/10.1103/physrevlett.92.255702) and the Python libraries ICET (2019),[19](https://doi.org/10.1002/adts.201900015) CLEASE (2019),[20](https://doi.org/10.1088/1361-648x/ab1bbc) and CASM (2022).[21](https://doi.org/10.1016/j.commatsci.2022.111897)\n\n## Variants\n\n**CVM versus generalized CE.** The 1984 paper also recast CVM as a self-consistency relation on renormalized cluster energies.[9](https://doi.org/10.1016/0378-4371%2884%2990096-7) The two differ in their variational parameters, cluster probabilities in CVM versus correlation functions in the generalized expansion.[2](https://par.nsf.gov/servlets/purl/10324364)\n\n**Basis variants.** The variable-basis cluster expansion (VBCE) uses basis functions whose expectation values vanish in the random state at the same concentration; for Mo-Ta, including concentration dependence of the interactions, explicitly or through volume, makes the expansion converge significantly faster than phenomenological Ising-like models.[10](https://link.springer.com/content/pdf/10.1007/s11669-017-0521-3.pdf) The mixed-space cluster expansion (MSCE) models short-ranged chemical interactions in real space and long-ranged strain interactions in reciprocal (k) space; originally limited to binary cubic single-sublattice alloys, it has been generalized to multiple sublattices and arbitrary crystal symmetry and is implemented in ATAT.[4](https://www.nature.com/articles/s41524-023-01029-0) Cluster expansions have also been reformulated for multicomponent ionic materials, handling charge-neutral configuration spaces, linear dependencies among correlation functions, and long-range electrostatics.[22](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.144202)\n\n**Fitting methods.** Setting the regularization matrices to zero gives ordinary least squares, which is prone to overfitting; ridge regression and LASSO follow from simple diagonal choices.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) Compressive sensing changed the field by allowing underdetermined problems, more candidate clusters than training structures, solved to sparse solutions with many zero ECIs.[8](https://icet.materialsmodeling.org/dev/get_started/cluster_expansions.html) In a comparison on 200 random Mo1-xVxCy structures, LASSO reached its validation-error minimum at about 35 nonzero parameters while ARDR achieved the same error with 20; the tutorial's authors recommend ARDR as the starting point for CE construction.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) A group-LASSO approach with hierarchical constraints built a 239-cluster model for the ternary Mo-V-Nb alloy from 800 training data with a lower cross-validation score than direct LASSO.[23](https://pubs.aip.org/aip/jcp/article/157/20/200901/2842100/Perspective-on-optimal-strategies-of-building)\n\n**Recent developments.** The embedded cluster expansion (eCE, 2024) simultaneously learns a low-dimensional embedding of site basis functions and the energy-model weights; a 2-eCE model of a six-component alloy needs on the order of \\( 10^{1} \\) features where the exact CE contains about \\( 10^{3} \\) descriptors, and the model extrapolates to element pairs absent from training.[24](https://arxiv.org/html/2409.06071v1) Graph-neural-network surrogates fine-tuned on a subset of DFT calculations now let thousands of structures be sampled cheaply before fitting.[12](https://arxiv.org/html/2403.18298) A nonlinear cluster expansion that adds machine-learning polynomial features to the standard basis resolved the longstanding failure for nonlinear composition dependence: with LASSO compressive sensing it exactly recovers a generating Redlich-Kister model, and a third-order nonlinear model with seven clusters outperformed standard models with up to 368 clusters for the clathrate band gap.[25](https://link.aps.org/doi/10.1103/68lv-86k6)\n\n## Applications\n\nCluster expansions coupled with [Monte Carlo sampling](https://www.edgepedia.org/monte-carlo-sampling) are an established technique for thermodynamic properties of multicomponent crystals, applied to metallic alloys, semiconductors, superionic conductors, battery electrodes, and surface catalysis.[5](https://www.nature.com/articles/s41524-024-01338-y) The tutorial literature lists phase-diagram prediction for metals and semiconductors, surfaces, nanoparticles, battery materials, perovskite photovoltaics, thermoelectrics, and high-entropy alloys, with extensions to activation barriers, vibrational properties, chemical expansion, and transport.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) Order–disorder transitions are a standard target: the CrCoNi ordering transition, with a heat-capacity peak around 940 K, is driven predominantly by nearest-neighbor pair interactions but substantially tuned by many-body interactions.[5](https://www.nature.com/articles/s41524-024-01338-y) Because CEs with 4-body terms describe alloy energetics to within the meV/atom range, quinary alloys can be fully described using the energetics of their constituent quaternaries alone.[26](https://link.springer.com/article/10.1007/s11669-015-0427-x)\n\n## Limitations and alternatives\n\n**Critical points.** The method falls short near critical points, where correlations become infinitely long; it is best suited to first-order transitions and to phases with substitutional disorder.[2](https://par.nsf.gov/servlets/purl/10324364)\n\n**Lattice relaxation.** Cluster expansions apply to ideal lattices; relaxed structures are mapped to the closest ideal-lattice occupation, so ECIs absorb relaxation and volume-change effects.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001) Systems with larger relaxation converge more slowly and may fail to converge altogether; using four Hamiltonians (first-principles, Lennard-Jones, Stillinger-Weber, embedded atom), a normalized mean-squared displacement below 0.1% usually generates a reliable model.[6](https://bsg.byu.edu/docs/papers/PhysRevB-96-014107.pdf) Long-ranged strain interactions require extension to reciprocal space, as in the MSCE.[4](https://www.nature.com/articles/s41524-023-01029-0) A single expansion also becomes difficult across a full composition range when charge states change with composition or species are aliovalent.[1](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001)\n\n**Truncation and fitting.** Truncating the infinite cluster sum requires careful testing, since many tiny long-range interactions can sum to large terms if the lattice contracts or expands differently across configurations.[7](https://indico.ictp.it/event/a12198/session/59/contribution/38/material/0/0.pdf) In Ni3V the expansion had failed unpredictably until an optimal truncation procedure produced agreement with measured values and predicted new low-energy structures.[18](https://doi.org/10.1103/physrevlett.92.255702) As linear regression, CE construction inherits the bias/variance dilemma; LASSO biases nonzero coefficients toward zero and can shrink large coefficients too severely.[23](https://pubs.aip.org/aip/jcp/article/157/20/200901/2842100/Perspective-on-optimal-strategies-of-building) The Fourier-transform cluster basis also fails to converge for properties with nonlinear concentration dependence, which the variable-basis expansion addresses.[11](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.224202)\n\n## References\n\n1. [Construction and Sampling of Alloy Cluster Expansions, A Tutorial (Ekborg-Tanner, Rosander, Fransson, Erhart, PRX Energy 3, 042001, published 17 October 2024)](https://link.aps.org/pdf/10.1103/PRXEnergy.3.042001)\n2. [Cluster Expansion of Alloy Theory: A Review of Historical Development and Modern Innovations](https://par.nsf.gov/servlets/purl/10324364)\n3. [ATAT Theoretical Background / user guide (van de Walle)](https://axelvandewalle.github.io/www-avdw/atat/manual/node19.html)\n4. [Generalization of the mixed-space cluster expansion method for arbitrary lattices (npj Computational Materials, 2023)](https://www.nature.com/articles/s41524-023-01029-0)\n5. [The cluster decomposition of the configurational energy of multicomponent alloys (npj Computational Materials, 2024)](https://www.nature.com/articles/s41524-024-01338-y)\n6. [Robustness of the cluster expansion: Assessing the roles of relaxation and numerical error (Phys. Rev. B 96, 014107)](https://bsg.byu.edu/docs/papers/PhysRevB-96-014107.pdf)\n7. [Cluster expansion tutorial (Blum, Hart, Richter, ICTP, 2011)](https://indico.ictp.it/event/a12198/session/59/contribution/38/material/0/0.pdf)\n8. [Cluster expansions, icet documentation](https://icet.materialsmodeling.org/dev/get_started/cluster_expansions.html)\n9. [Generalized cluster description of multicomponent systems (Physica A Statistical Mechanics and its Applications, 1984)](https://doi.org/10.1016/0378-4371%2884%2990096-7)\n10. [Foundations and Practical Implementations of the Cluster Expansion (Journal of Phase Equilibria and Diffusion, 2017, DOI 10.1007/s11669-017-0521-3)](https://link.springer.com/content/pdf/10.1007/s11669-017-0521-3.pdf)\n11. [Cluster expansion and the configurational theory of alloys (J. M. Sanchez, Phys. Rev. B 81, 224202, 2010)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.224202)\n12. [Deciphering Chemical Ordering in High Entropy Materials: A Machine Learning-Accelerated High-throughput Cluster Expansion Approach (arXiv:2403.18298, 2024)](https://arxiv.org/html/2403.18298)\n13. [Ryoichi Kikuchi (1951). A Theory of Cooperative Phenomena. Physical Review.](https://doi.org/10.1103/physrev.81.988)\n14. [J. M. Sanchez (1993). Cluster expansions and the configurational energy of alloys. Physical review. B, Condensed matter.](https://doi.org/10.1103/physrevb.48.14013)\n15. [Tim Mueller, Gerbrand Ceder (2009). Bayesian approach to cluster expansions. Physical Review B.](https://doi.org/10.1103/physrevb.80.024103)\n16. [Lance J. Nelson and colleagues (2013). Cluster expansion made easy with Bayesian compressive sensing. Physical Review B.](https://doi.org/10.1103/physrevb.88.155105)\n17. [Alex Zunger and colleagues (2002). Obtaining Ising-like expansions for binary alloys from first principles. Modelling and Simulation in Materials Science and Engineering.](https://doi.org/10.1088/0965-0393/10/6/306)\n18. [Nikolai A. Zarkevich, D. D. Johnson (2004). Reliable First-Principles Alloy Thermodynamics via Truncated Cluster Expansions. Physical Review Letters.](https://doi.org/10.1103/physrevlett.92.255702)\n19. [Mattias Ångqvist and colleagues (2019). ICET – A Python Library for Constructing and Sampling Alloy Cluster Expansions. Advanced Theory and Simulations.](https://doi.org/10.1002/adts.201900015)\n20. [Jin Hyun Chang and colleagues (2019). CLEASE: a versatile and user-friendly implementation of cluster expansion method. Journal of Physics Condensed Matter.](https://doi.org/10.1088/1361-648x/ab1bbc)\n21. [Brian Puchala and colleagues (2022). CASM, A software package for first-principles based study of multicomponent crystalline solids. Computational Materials Science.](https://doi.org/10.1016/j.commatsci.2022.111897)\n22. [Cluster expansions of multicomponent ionic materials: Formalism and methodology (Phys. Rev. B 106, 144202, 2022)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.144202)\n23. [Perspective on optimal strategies of building cluster expansion models for configurationally disordered materials (J. Chem. Phys. 157, 200901, 2022)](https://pubs.aip.org/aip/jcp/article/157/20/200901/2842100/Perspective-on-optimal-strategies-of-building)\n24. [Constructing multicomponent cluster expansions with machine-learning and chemical embedding (arXiv:2409.06071, 2024; published-version excerpts merged from EPFL infoscience copy)](https://arxiv.org/html/2409.06071v1)\n25. [Cluster expansion toward nonlinear modeling and classification (Physical Review Research)](https://link.aps.org/doi/10.1103/68lv-86k6)\n26. [Cluster Expansions for Thermodynamics and Kinetics of Multicomponent Alloys (Journal of Phase Equilibria and Diffusion, 2015)](https://link.springer.com/article/10.1007/s11669-015-0427-x)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Condensed matter physics › Crystal and structural condensed matter*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 25, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/cluster-expansion-statistical-mechanics", "markdown_url": "https://www.edgepedia.org/cluster-expansion-statistical-mechanics.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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Edgepedia Community License 1.0.", "speakable": "A cluster expansion expresses a lattice configuration's energy as a sum over cluster correlation functions, letting alloy thermodynamics be sampled after training on few structures." }