Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain Article Swipe
Debraj Das
,
Sushanta Dattagupta
,
Shamik Gupta
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.1088/1742-5468/ac6256
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.1088/1742-5468/ac6256
In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. The formalism is very general in that it applies to any quantum system, to any form of non-unitary interaction, and to any probability distribution for the random times. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, e.g. in modelling nano wires. Considering the case of one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, owing to spontaneous quantum fluctuations tunnels between the nearest-neighbour sites. We consider two representative forms of interactions, one that implements a stochastic reset of quantum dynamics in which the density operator is at random times reset to its initial form, and one in which projective measurements are performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement (MSD) of the particle about its initial location. This stands in stark contrast to the behavior in the absence of interactions, when the particle has an unbounded growth of the MSD in time, with no signatures of localization. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which it does not require us to perform measurements at exactly regular intervals that are hallmarks of such scenarios.
Related Topics
Concepts
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1088/1742-5468/ac6256
- OA Status
- hybrid
- Cited By
- 21
- References
- 77
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4229373476
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4229373476Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1088/1742-5468/ac6256Digital Object Identifier
- Title
-
Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chainWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-05-01Full publication date if available
- Authors
-
Debraj Das, Sushanta Dattagupta, Shamik GuptaList of authors in order
- Landing page
-
https://doi.org/10.1088/1742-5468/ac6256Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
hybridOpen access status per OpenAlex
- OA URL
-
https://doi.org/10.1088/1742-5468/ac6256Direct OA link when available
- Concepts
-
Randomness, Statistical physics, Quantum, Physics, Unitary state, Independent and identically distributed random variables, Quantum mechanics, Mathematics, Random variable, Statistics, Law, Political scienceTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
21Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 8, 2024: 3, 2023: 6, 2022: 4Per-year citation counts (last 5 years)
- References (count)
-
77Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4229373476 |
|---|---|
| doi | https://doi.org/10.1088/1742-5468/ac6256 |
| ids.doi | https://doi.org/10.1088/1742-5468/ac6256 |
| ids.openalex | https://openalex.org/W4229373476 |
| fwci | 12.32546793 |
| type | article |
| title | Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain |
| biblio.issue | 5 |
| biblio.volume | 2022 |
| biblio.last_page | 053101 |
| biblio.first_page | 053101 |
| topics[0].id | https://openalex.org/T11177 |
| topics[0].field.id | https://openalex.org/fields/31 |
| topics[0].field.display_name | Physics and Astronomy |
| topics[0].score | 0.998199999332428 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/3107 |
| topics[0].subfield.display_name | Atomic and Molecular Physics, and Optics |
| topics[0].display_name | Spectroscopy and Quantum Chemical Studies |
| topics[1].id | https://openalex.org/T13187 |
| topics[1].field.id | https://openalex.org/fields/13 |
| topics[1].field.display_name | Biochemistry, Genetics and Molecular Biology |
| topics[1].score | 0.9976999759674072 |
| topics[1].domain.id | https://openalex.org/domains/1 |
| topics[1].domain.display_name | Life Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1312 |
| topics[1].subfield.display_name | Molecular Biology |
| topics[1].display_name | Diffusion and Search Dynamics |
| topics[2].id | https://openalex.org/T12261 |
| topics[2].field.id | https://openalex.org/fields/31 |
| topics[2].field.display_name | Physics and Astronomy |
| topics[2].score | 0.9943000078201294 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/3109 |
| topics[2].subfield.display_name | Statistical and Nonlinear Physics |
| topics[2].display_name | Statistical Mechanics and Entropy |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C125112378 |
| concepts[0].level | 2 |
| concepts[0].score | 0.7219029068946838 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q176640 |
| concepts[0].display_name | Randomness |
| concepts[1].id | https://openalex.org/C121864883 |
| concepts[1].level | 1 |
| concepts[1].score | 0.6467053890228271 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q677916 |
| concepts[1].display_name | Statistical physics |
| concepts[2].id | https://openalex.org/C84114770 |
| concepts[2].level | 2 |
| concepts[2].score | 0.498333215713501 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q46344 |
| concepts[2].display_name | Quantum |
| concepts[3].id | https://openalex.org/C121332964 |
| concepts[3].level | 0 |
| concepts[3].score | 0.4918692111968994 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[3].display_name | Physics |
| concepts[4].id | https://openalex.org/C67820243 |
| concepts[4].level | 2 |
| concepts[4].score | 0.48216456174850464 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q179164 |
| concepts[4].display_name | Unitary state |
| concepts[5].id | https://openalex.org/C141513077 |
| concepts[5].level | 3 |
| concepts[5].score | 0.43276768922805786 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q378542 |
| concepts[5].display_name | Independent and identically distributed random variables |
| concepts[6].id | https://openalex.org/C62520636 |
| concepts[6].level | 1 |
| concepts[6].score | 0.39208608865737915 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[6].display_name | Quantum mechanics |
| concepts[7].id | https://openalex.org/C33923547 |
| concepts[7].level | 0 |
| concepts[7].score | 0.36789482831954956 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[7].display_name | Mathematics |
| concepts[8].id | https://openalex.org/C122123141 |
| concepts[8].level | 2 |
| concepts[8].score | 0.26118844747543335 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q176623 |
| concepts[8].display_name | Random variable |
| concepts[9].id | https://openalex.org/C105795698 |
| concepts[9].level | 1 |
| concepts[9].score | 0.11697036027908325 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q12483 |
| concepts[9].display_name | Statistics |
| concepts[10].id | https://openalex.org/C199539241 |
| concepts[10].level | 1 |
| concepts[10].score | 0.0 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q7748 |
| concepts[10].display_name | Law |
| concepts[11].id | https://openalex.org/C17744445 |
| concepts[11].level | 0 |
| concepts[11].score | 0.0 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q36442 |
| concepts[11].display_name | Political science |
| keywords[0].id | https://openalex.org/keywords/randomness |
| keywords[0].score | 0.7219029068946838 |
| keywords[0].display_name | Randomness |
| keywords[1].id | https://openalex.org/keywords/statistical-physics |
| keywords[1].score | 0.6467053890228271 |
| keywords[1].display_name | Statistical physics |
| keywords[2].id | https://openalex.org/keywords/quantum |
| keywords[2].score | 0.498333215713501 |
| keywords[2].display_name | Quantum |
| keywords[3].id | https://openalex.org/keywords/physics |
| keywords[3].score | 0.4918692111968994 |
| keywords[3].display_name | Physics |
| keywords[4].id | https://openalex.org/keywords/unitary-state |
| keywords[4].score | 0.48216456174850464 |
| keywords[4].display_name | Unitary state |
| keywords[5].id | https://openalex.org/keywords/independent-and-identically-distributed-random-variables |
| keywords[5].score | 0.43276768922805786 |
| keywords[5].display_name | Independent and identically distributed random variables |
| keywords[6].id | https://openalex.org/keywords/quantum-mechanics |
| keywords[6].score | 0.39208608865737915 |
| keywords[6].display_name | Quantum mechanics |
| keywords[7].id | https://openalex.org/keywords/mathematics |
| keywords[7].score | 0.36789482831954956 |
| keywords[7].display_name | Mathematics |
| keywords[8].id | https://openalex.org/keywords/random-variable |
| keywords[8].score | 0.26118844747543335 |
| keywords[8].display_name | Random variable |
| keywords[9].id | https://openalex.org/keywords/statistics |
| keywords[9].score | 0.11697036027908325 |
| keywords[9].display_name | Statistics |
| language | en |
| locations[0].id | doi:10.1088/1742-5468/ac6256 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S167943036 |
| locations[0].source.issn | 1742-5468 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | False |
| locations[0].source.issn_l | 1742-5468 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Journal of Statistical Mechanics Theory and Experiment |
| locations[0].source.host_organization | https://openalex.org/P4310311669 |
| locations[0].source.host_organization_name | Institute of Physics |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310311669 |
| locations[0].source.host_organization_lineage_names | Institute of Physics |
| locations[0].license | cc-by |
| locations[0].pdf_url | |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Journal of Statistical Mechanics: Theory and Experiment |
| locations[0].landing_page_url | https://doi.org/10.1088/1742-5468/ac6256 |
| locations[1].id | pmh:oai:arXiv.org:2106.14181 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | |
| locations[1].pdf_url | https://arxiv.org/pdf/2106.14181 |
| locations[1].version | submittedVersion |
| locations[1].raw_type | text |
| locations[1].license_id | |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | http://arxiv.org/abs/2106.14181 |
| indexed_in | arxiv, crossref |
| authorships[0].author.id | https://openalex.org/A5065965915 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-8776-7065 |
| authorships[0].author.display_name | Debraj Das |
| authorships[0].countries | GB |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I36234482 |
| authorships[0].affiliations[0].raw_affiliation_string | Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom |
| authorships[0].institutions[0].id | https://openalex.org/I36234482 |
| authorships[0].institutions[0].ror | https://ror.org/0524sp257 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I36234482 |
| authorships[0].institutions[0].country_code | GB |
| authorships[0].institutions[0].display_name | University of Bristol |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Debraj Das |
| authorships[0].is_corresponding | True |
| authorships[0].raw_affiliation_strings | Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom |
| authorships[1].author.id | https://openalex.org/A5037648326 |
| authorships[1].author.orcid | https://orcid.org/0000-0002-9857-5843 |
| authorships[1].author.display_name | Sushanta Dattagupta |
| authorships[1].countries | IN |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I152754861 |
| authorships[1].affiliations[0].raw_affiliation_string | Bose Institute, Kolkata 700054, India |
| authorships[1].institutions[0].id | https://openalex.org/I152754861 |
| authorships[1].institutions[0].ror | https://ror.org/01a5mqy88 |
| authorships[1].institutions[0].type | facility |
| authorships[1].institutions[0].lineage | https://openalex.org/I152754861 |
| authorships[1].institutions[0].country_code | IN |
| authorships[1].institutions[0].display_name | Bose Institute |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Sushanta Dattagupta |
| authorships[1].is_corresponding | True |
| authorships[1].raw_affiliation_strings | Bose Institute, Kolkata 700054, India |
| authorships[2].author.id | https://openalex.org/A5086347609 |
| authorships[2].author.orcid | https://orcid.org/0000-0002-6080-4890 |
| authorships[2].author.display_name | Shamik Gupta |
| authorships[2].countries | IN, IT |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I12323705 |
| authorships[2].affiliations[0].raw_affiliation_string | Quantitative Life Sciences Section, ICTP—Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy |
| authorships[2].affiliations[1].institution_ids | https://openalex.org/I2799306724, https://openalex.org/I34034455 |
| authorships[2].affiliations[1].raw_affiliation_string | Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India |
| authorships[2].institutions[0].id | https://openalex.org/I2799306724 |
| authorships[2].institutions[0].ror | https://ror.org/019wbnq12 |
| authorships[2].institutions[0].type | other |
| authorships[2].institutions[0].lineage | https://openalex.org/I2799306724 |
| authorships[2].institutions[0].country_code | IN |
| authorships[2].institutions[0].display_name | Ramakrishna Mission Vidyamandira |
| authorships[2].institutions[1].id | https://openalex.org/I34034455 |
| authorships[2].institutions[1].ror | https://ror.org/03kp2qt98 |
| authorships[2].institutions[1].type | education |
| authorships[2].institutions[1].lineage | https://openalex.org/I34034455 |
| authorships[2].institutions[1].country_code | IN |
| authorships[2].institutions[1].display_name | Ramakrishna Mission Vivekananda Educational and Research Institute |
| authorships[2].institutions[2].id | https://openalex.org/I12323705 |
| authorships[2].institutions[2].ror | https://ror.org/009gyvm78 |
| authorships[2].institutions[2].type | facility |
| authorships[2].institutions[2].lineage | https://openalex.org/I12323705, https://openalex.org/I1286959531, https://openalex.org/I1293226324, https://openalex.org/I2801247003 |
| authorships[2].institutions[2].country_code | IT |
| authorships[2].institutions[2].display_name | The Abdus Salam International Centre for Theoretical Physics (ICTP) |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Shamik Gupta |
| authorships[2].is_corresponding | True |
| authorships[2].raw_affiliation_strings | Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India, Quantitative Life Sciences Section, ICTP—Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://doi.org/10.1088/1742-5468/ac6256 |
| open_access.oa_status | hybrid |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T11177 |
| primary_topic.field.id | https://openalex.org/fields/31 |
| primary_topic.field.display_name | Physics and Astronomy |
| primary_topic.score | 0.998199999332428 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/3107 |
| primary_topic.subfield.display_name | Atomic and Molecular Physics, and Optics |
| primary_topic.display_name | Spectroscopy and Quantum Chemical Studies |
| related_works | https://openalex.org/W3034924094, https://openalex.org/W3094954546, https://openalex.org/W1488708774, https://openalex.org/W1982811510, https://openalex.org/W4391100477, https://openalex.org/W2402189625, https://openalex.org/W4327779705, https://openalex.org/W1513698804, https://openalex.org/W4310560702, https://openalex.org/W2029712093 |
| cited_by_count | 21 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 8 |
| counts_by_year[1].year | 2024 |
| counts_by_year[1].cited_by_count | 3 |
| counts_by_year[2].year | 2023 |
| counts_by_year[2].cited_by_count | 6 |
| counts_by_year[3].year | 2022 |
| counts_by_year[3].cited_by_count | 4 |
| locations_count | 2 |
| best_oa_location.id | doi:10.1088/1742-5468/ac6256 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S167943036 |
| best_oa_location.source.issn | 1742-5468 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | False |
| best_oa_location.source.issn_l | 1742-5468 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Journal of Statistical Mechanics Theory and Experiment |
| best_oa_location.source.host_organization | https://openalex.org/P4310311669 |
| best_oa_location.source.host_organization_name | Institute of Physics |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310311669 |
| best_oa_location.source.host_organization_lineage_names | Institute of Physics |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Journal of Statistical Mechanics: Theory and Experiment |
| best_oa_location.landing_page_url | https://doi.org/10.1088/1742-5468/ac6256 |
| primary_location.id | doi:10.1088/1742-5468/ac6256 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S167943036 |
| primary_location.source.issn | 1742-5468 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | False |
| primary_location.source.issn_l | 1742-5468 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Journal of Statistical Mechanics Theory and Experiment |
| primary_location.source.host_organization | https://openalex.org/P4310311669 |
| primary_location.source.host_organization_name | Institute of Physics |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310311669 |
| primary_location.source.host_organization_lineage_names | Institute of Physics |
| primary_location.license | cc-by |
| primary_location.pdf_url | |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Journal of Statistical Mechanics: Theory and Experiment |
| primary_location.landing_page_url | https://doi.org/10.1088/1742-5468/ac6256 |
| publication_date | 2022-05-01 |
| publication_year | 2022 |
| referenced_works | https://openalex.org/W2071093123, https://openalex.org/W2051335738, https://openalex.org/W1969211920, https://openalex.org/W2006971528, https://openalex.org/W2085247228, https://openalex.org/W4224228578, https://openalex.org/W2807650876, https://openalex.org/W2013879218, https://openalex.org/W2593272000, https://openalex.org/W1965304117, https://openalex.org/W1994557753, https://openalex.org/W2566076131, https://openalex.org/W1989941945, https://openalex.org/W1967200406, https://openalex.org/W1977079737, https://openalex.org/W2166743586, https://openalex.org/W2320235279, https://openalex.org/W2088276743, https://openalex.org/W2414452494, https://openalex.org/W2046660045, https://openalex.org/W2242190300, https://openalex.org/W2053742402, https://openalex.org/W2344580653, https://openalex.org/W2027872720, https://openalex.org/W2980928409, https://openalex.org/W2118507995, https://openalex.org/W3016430050, https://openalex.org/W6793055034, https://openalex.org/W3087946577, https://openalex.org/W2185436178, https://openalex.org/W2486160458, https://openalex.org/W2343391028, https://openalex.org/W2799155248, https://openalex.org/W2963455530, https://openalex.org/W2019414322, https://openalex.org/W1582734385, https://openalex.org/W2300091764, https://openalex.org/W2552294309, https://openalex.org/W2766282894, https://openalex.org/W6767786530, https://openalex.org/W2748213412, https://openalex.org/W2921876320, https://openalex.org/W2921673115, https://openalex.org/W3081179211, https://openalex.org/W2981753498, https://openalex.org/W3107821816, https://openalex.org/W3140230075, https://openalex.org/W6794277167, https://openalex.org/W3110478206, https://openalex.org/W1977392168, https://openalex.org/W1564175003, https://openalex.org/W1483893079, https://openalex.org/W2057908063, https://openalex.org/W4299929875, https://openalex.org/W3098596919, https://openalex.org/W4254398057, https://openalex.org/W3104813263, https://openalex.org/W3105146067, https://openalex.org/W3105473114, https://openalex.org/W561029282, https://openalex.org/W3100582382, https://openalex.org/W4230959423, https://openalex.org/W3106408390, https://openalex.org/W3156678874, https://openalex.org/W197423884, https://openalex.org/W3147017318, https://openalex.org/W3089116201, https://openalex.org/W3102937562, https://openalex.org/W3098395760, https://openalex.org/W2963266352, https://openalex.org/W3122516470, https://openalex.org/W3103066384, https://openalex.org/W3103892560, https://openalex.org/W2971938676, https://openalex.org/W420010159, https://openalex.org/W3163106977, https://openalex.org/W3010636260 |
| referenced_works_count | 77 |
| abstract_inverted_index.a | 8, 28, 33, 151, 158, 191, 248 |
| abstract_inverted_index.In | 1, 225, 290 |
| abstract_inverted_index.We | 109, 181 |
| abstract_inverted_index.an | 277, 313 |
| abstract_inverted_index.at | 13, 46, 203, 222, 243, 296, 357 |
| abstract_inverted_index.be | 75, 327 |
| abstract_inverted_index.in | 86, 117, 126, 129, 133, 197, 213, 262, 268, 283, 312, 320, 342 |
| abstract_inverted_index.is | 55, 83, 163, 202, 238, 337 |
| abstract_inverted_index.it | 88, 349 |
| abstract_inverted_index.no | 286 |
| abstract_inverted_index.of | 4, 7, 43, 97, 114, 120, 140, 150, 157, 186, 194, 253, 271, 280, 288, 293, 308, 316, 323, 335, 364 |
| abstract_inverted_index.on | 168, 219, 240, 329 |
| abstract_inverted_index.or | 27 |
| abstract_inverted_index.to | 20, 37, 58, 74, 90, 94, 101, 172, 207, 247, 265, 304, 326, 339, 354 |
| abstract_inverted_index.us | 353 |
| abstract_inverted_index.we | 31, 72, 229, 299 |
| abstract_inverted_index.MSD | 282 |
| abstract_inverted_index.The | 53, 81, 333 |
| abstract_inverted_index.and | 77, 100, 211 |
| abstract_inverted_index.any | 47, 91, 95, 102 |
| abstract_inverted_index.are | 217, 362 |
| abstract_inverted_index.but | 347 |
| abstract_inverted_index.due | 19 |
| abstract_inverted_index.for | 105, 164 |
| abstract_inverted_index.has | 276 |
| abstract_inverted_index.how | 235 |
| abstract_inverted_index.its | 208, 257 |
| abstract_inverted_index.not | 351 |
| abstract_inverted_index.one | 141, 188, 212, 341 |
| abstract_inverted_index.our | 232 |
| abstract_inverted_index.the | 2, 24, 39, 44, 50, 59, 64, 106, 115, 118, 121, 138, 143, 148, 155, 161, 169, 178, 199, 220, 226, 236, 241, 254, 266, 269, 274, 281, 291, 305, 309, 317, 321, 324, 330, 340 |
| abstract_inverted_index.two | 111, 183 |
| abstract_inverted_index.This | 260 |
| abstract_inverted_index.Zeno | 344 |
| abstract_inverted_index.case | 139, 292 |
| abstract_inverted_index.does | 350 |
| abstract_inverted_index.e.g. | 132 |
| abstract_inverted_index.form | 96 |
| abstract_inverted_index.long | 244 |
| abstract_inverted_index.most | 165 |
| abstract_inverted_index.nano | 135 |
| abstract_inverted_index.show | 300 |
| abstract_inverted_index.such | 365 |
| abstract_inverted_index.that | 87, 189, 301, 361 |
| abstract_inverted_index.time | 48, 66 |
| abstract_inverted_index.very | 84 |
| abstract_inverted_index.when | 273 |
| abstract_inverted_index.with | 16, 22, 56, 63, 231, 285 |
| abstract_inverted_index.(MSD) | 252 |
| abstract_inverted_index.about | 256 |
| abstract_inverted_index.case, | 228 |
| abstract_inverted_index.chain | 146 |
| abstract_inverted_index.decay | 319 |
| abstract_inverted_index.exact | 233 |
| abstract_inverted_index.form, | 210 |
| abstract_inverted_index.forms | 185 |
| abstract_inverted_index.found | 328 |
| abstract_inverted_index.model | 124 |
| abstract_inverted_index.owing | 171 |
| abstract_inverted_index.reset | 193, 206 |
| abstract_inverted_index.sites | 156, 242 |
| abstract_inverted_index.stark | 263 |
| abstract_inverted_index.state | 307 |
| abstract_inverted_index.time, | 284 |
| abstract_inverted_index.times | 15, 166, 205 |
| abstract_inverted_index.which | 71, 198, 214, 348 |
| abstract_inverted_index.adduce | 32 |
| abstract_inverted_index.amount | 334 |
| abstract_inverted_index.during | 49 |
| abstract_inverted_index.effect | 345 |
| abstract_inverted_index.either | 23 |
| abstract_inverted_index.former | 227 |
| abstract_inverted_index.growth | 279 |
| abstract_inverted_index.models | 147 |
| abstract_inverted_index.motion | 149 |
| abstract_inverted_index.obtain | 38 |
| abstract_inverted_index.random | 14, 65, 79, 107, 204, 223, 297 |
| abstract_inverted_index.sites, | 170 |
| abstract_inverted_index.sites. | 180 |
| abstract_inverted_index.stands | 261 |
| abstract_inverted_index.state. | 332 |
| abstract_inverted_index.system | 11, 45, 221 |
| abstract_inverted_index.times, | 245, 298 |
| abstract_inverted_index.times. | 108, 224 |
| abstract_inverted_index.wires. | 136 |
| abstract_inverted_index.absence | 270 |
| abstract_inverted_index.applies | 89 |
| abstract_inverted_index.average | 40, 54 |
| abstract_inverted_index.between | 68, 154, 177 |
| abstract_inverted_index.charged | 152 |
| abstract_inverted_index.context | 3, 119 |
| abstract_inverted_index.density | 41, 200 |
| abstract_inverted_index.exactly | 358 |
| abstract_inverted_index.general | 34, 85 |
| abstract_inverted_index.generic | 9 |
| abstract_inverted_index.initial | 209, 258, 306, 331 |
| abstract_inverted_index.leading | 246 |
| abstract_inverted_index.perform | 355 |
| abstract_inverted_index.provide | 110 |
| abstract_inverted_index.quantum | 10, 92, 174, 195 |
| abstract_inverted_index.regular | 359 |
| abstract_inverted_index.require | 352 |
| abstract_inverted_index.respect | 57 |
| abstract_inverted_index.results | 234, 311 |
| abstract_inverted_index.system, | 93 |
| abstract_inverted_index.tunnels | 176 |
| abstract_inverted_index.unitary | 5 |
| abstract_inverted_index.various | 127 |
| abstract_inverted_index.wherein | 160 |
| abstract_inverted_index.Abstract | 0 |
| abstract_inverted_index.behavior | 267 |
| abstract_inverted_index.consider | 73, 182 |
| abstract_inverted_index.contexts | 128 |
| abstract_inverted_index.contrast | 264 |
| abstract_inverted_index.dynamics | 196 |
| abstract_inverted_index.explicit | 112 |
| abstract_inverted_index.external | 25 |
| abstract_inverted_index.lattice, | 159 |
| abstract_inverted_index.operator | 42, 201 |
| abstract_inverted_index.particle | 153, 162, 237, 255, 275, 310, 325 |
| abstract_inverted_index.physics, | 131 |
| abstract_inverted_index.relevant | 125 |
| abstract_inverted_index.repeated | 302 |
| abstract_inverted_index.temporal | 318 |
| abstract_inverted_index.classical | 60 |
| abstract_inverted_index.dynamical | 51 |
| abstract_inverted_index.effective | 314 |
| abstract_inverted_index.evolution | 6, 18 |
| abstract_inverted_index.formalism | 82, 116 |
| abstract_inverted_index.framework | 36 |
| abstract_inverted_index.hallmarks | 363 |
| abstract_inverted_index.intervals | 67, 360 |
| abstract_inverted_index.localized | 167, 239 |
| abstract_inverted_index.location. | 259 |
| abstract_inverted_index.measuring | 29 |
| abstract_inverted_index.modelling | 134 |
| abstract_inverted_index.performed | 218 |
| abstract_inverted_index.so-called | 122 |
| abstract_inverted_index.unbounded | 278 |
| abstract_inverted_index.apparatus, | 30 |
| abstract_inverted_index.associated | 62 |
| abstract_inverted_index.comparable | 338 |
| abstract_inverted_index.dimension, | 142 |
| abstract_inverted_index.evolution. | 52 |
| abstract_inverted_index.implements | 190 |
| abstract_inverted_index.projection | 303 |
| abstract_inverted_index.projective | 215, 294 |
| abstract_inverted_index.randomness | 61 |
| abstract_inverted_index.scenarios, | 346 |
| abstract_inverted_index.scenarios. | 366 |
| abstract_inverted_index.signatures | 287 |
| abstract_inverted_index.stochastic | 192 |
| abstract_inverted_index.successive | 69 |
| abstract_inverted_index.variables. | 80 |
| abstract_inverted_index.Considering | 137 |
| abstract_inverted_index.demonstrate | 230 |
| abstract_inverted_index.environment | 26 |
| abstract_inverted_index.independent | 76 |
| abstract_inverted_index.interrupted | 12 |
| abstract_inverted_index.non-unitary | 17, 98 |
| abstract_inverted_index.probability | 103, 322 |
| abstract_inverted_index.solid-state | 130 |
| abstract_inverted_index.spontaneous | 173 |
| abstract_inverted_index.suppression | 315, 336 |
| abstract_inverted_index.theoretical | 35 |
| abstract_inverted_index.applications | 113 |
| abstract_inverted_index.conventional | 343 |
| abstract_inverted_index.displacement | 251 |
| abstract_inverted_index.distribution | 104 |
| abstract_inverted_index.fluctuations | 175 |
| abstract_inverted_index.interaction, | 99 |
| abstract_inverted_index.interactions | 21 |
| abstract_inverted_index.mean-squared | 250 |
| abstract_inverted_index.measurements | 216, 295, 356 |
| abstract_inverted_index.corresponding | 144 |
| abstract_inverted_index.interactions, | 70, 187, 272 |
| abstract_inverted_index.localization. | 289 |
| abstract_inverted_index.tight-binding | 123, 145 |
| abstract_inverted_index.representative | 184 |
| abstract_inverted_index.time-independent | 249 |
| abstract_inverted_index.nearest-neighbour | 179 |
| abstract_inverted_index.identically-distributed | 78 |
| cited_by_percentile_year.max | 99 |
| cited_by_percentile_year.min | 96 |
| corresponding_author_ids | https://openalex.org/A5065965915, https://openalex.org/A5086347609, https://openalex.org/A5037648326 |
| countries_distinct_count | 3 |
| institutions_distinct_count | 3 |
| corresponding_institution_ids | https://openalex.org/I12323705, https://openalex.org/I152754861, https://openalex.org/I2799306724, https://openalex.org/I34034455, https://openalex.org/I36234482 |
| citation_normalized_percentile.value | 0.9354482 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |