Thermodynamic Consistency of the Cushman Method of Computing the Configurational Entropy of a Landscape Lattice

There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the thermodynamic consistency of these approaches. I posit that for any of these methods to be fully thermodynamically consistent, they must meet th...

Full description

Saved in:
Bibliographic Details
Published inEntropy (Basel, Switzerland) Vol. 23; no. 11; p. 1420
Main Author Cushman, Samuel A
Format Journal Article
LanguageEnglish
Published Switzerland MDPI AG 28.10.2021
MDPI
Subjects
Online AccessGet full text

Cover

Loading…
Abstract There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the thermodynamic consistency of these approaches. I posit that for any of these methods to be fully thermodynamically consistent, they must meet three conditions. First, the computed entropies must lie along the theoretical distribution of entropies as a function of total edge length, which Cushman showed was a parabolic function following from the fact that there is a normal distribution of permuted edge lengths, the entropy is the logarithm of the number of microstates in a macrostate, and the logarithm of a normal distribution is a parabolic function. Second, the entropy must increase over time through the period of the random mixing simulation, following the expectation that entropy increases in a closed system. Third, at full mixing, the entropy will fluctuate randomly around the maximum theoretical value, associated with a perfectly random arrangement of the lattice. I evaluated these criteria in a test condition involving a binary, two-class landscape using the Cushman method of directly applying the Boltzmann relation (s = klogW) to permuted landscape configurations and measuring the distribution of total edge length. The results show that the Cushman method directly applying the classical Boltzmann relation is fully consistent with these criteria and therefore fully thermodynamically consistent. I suggest that this method, which is a direct application of the classical and iconic formulation of Boltzmann, has advantages given its direct interpretability, theoretical elegance, and thermodynamic consistency.
AbstractList There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the thermodynamic consistency of these approaches. I posit that for any of these methods to be fully thermodynamically consistent, they must meet three conditions. First, the computed entropies must lie along the theoretical distribution of entropies as a function of total edge length, which Cushman showed was a parabolic function following from the fact that there is a normal distribution of permuted edge lengths, the entropy is the logarithm of the number of microstates in a macrostate, and the logarithm of a normal distribution is a parabolic function. Second, the entropy must increase over time through the period of the random mixing simulation, following the expectation that entropy increases in a closed system. Third, at full mixing, the entropy will fluctuate randomly around the maximum theoretical value, associated with a perfectly random arrangement of the lattice. I evaluated these criteria in a test condition involving a binary, two-class landscape using the Cushman method of directly applying the Boltzmann relation (s = klogW) to permuted landscape configurations and measuring the distribution of total edge length. The results show that the Cushman method directly applying the classical Boltzmann relation is fully consistent with these criteria and therefore fully thermodynamically consistent. I suggest that this method, which is a direct application of the classical and iconic formulation of Boltzmann, has advantages given its direct interpretability, theoretical elegance, and thermodynamic consistency.
Author Cushman, Samuel A
AuthorAffiliation USDA Forest Service, Rocky Mountain Research Station, Flagstaff, AZ 86004, USA; Samuel.cushman@usda.gov
AuthorAffiliation_xml – name: USDA Forest Service, Rocky Mountain Research Station, Flagstaff, AZ 86004, USA; Samuel.cushman@usda.gov
Author_xml – sequence: 1
  givenname: Samuel A
  surname: Cushman
  fullname: Cushman, Samuel A
  organization: USDA Forest Service, Rocky Mountain Research Station, Flagstaff, AZ 86004, USA
BackLink https://www.ncbi.nlm.nih.gov/pubmed/34828118$$D View this record in MEDLINE/PubMed
BookMark eNpdkktv1DAUhS1URB-w4A-gSGzoYsCvOPYGCUWlVBrEpqwtx7mZeJTYg-1Umn-PO1NGLStf3fv56Fz7XKIzHzwg9J7gz4wp_AUoI4Rwil-hC4KVWnGG8dmz-hxdprTFmDJKxBt0zrikkhB5gcL9CHEO_d6b2dmqDT65lMHbfRWGKo9QtUsaZ-Orn5DH0D922zDvluz85jgPfnCbJZrsgjdTdeNzDLvDdVOtje-TNTsoVc7Owlv0ejBTgndP5xX6_f3mvv2xWv-6vWu_rVeWC5VXHWZ1R8nAG6wIGOCqGYyhqpcELDZEqLKu5N1QE0u4hEE2TOCaEcDUMg7sCt0ddftgtnoX3WziXgfj9KER4kabWAxNoBlVUGOiGKOW96KWTAnFjWF9z4EOXdH6etTaLd0MvYWyoZleiL6ceDfqTXjQUpBGUFoEPj0JxPBngZT17JKFaTIewpI0FZiXv2mEKOjH_9BtWGJ51wNFMWMS40JdHykbQ0oRhpMZgvVjJPQpEoX98Nz9ifyXAfYXG0yyMw
CitedBy_id crossref_primary_10_1007_s10980_022_01513_w
crossref_primary_10_3390_e23121616
crossref_primary_10_3389_fgene_2023_1269792
crossref_primary_10_3390_e25030405
crossref_primary_10_3390_e25121653
crossref_primary_10_3390_e23111425
crossref_primary_10_1007_s10980_022_01554_1
Cites_doi 10.1007/s10980-019-00876-x
10.1002/j.1538-7305.1948.tb01338.x
10.3390/e22090937
10.1017/CBO9780511614415
10.1007/s13752-014-0162-2
10.1007/s10980-014-0108-x
10.3389/fevo.2019.00440
10.3390/e23111425
10.1007/s10980-020-01177-4
10.3390/e23121616
10.1021/i160042a023
10.3390/e20040298
10.3390/e22040381
10.1007/s10980-015-0305-2
10.1111/tgis.12315
10.1007/s10980-019-00814-x
10.1002/cplx.20388
ContentType Journal Article
Copyright 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
2021 by the author. 2021
Copyright_xml – notice: 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
– notice: 2021 by the author. 2021
DBID NPM
AAYXX
CITATION
7TB
8FD
8FE
8FG
ABJCF
ABUWG
AFKRA
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
FR3
HCIFZ
KR7
L6V
M7S
PIMPY
PQEST
PQQKQ
PQUKI
PRINS
PTHSS
7X8
5PM
DOA
DOI 10.3390/e23111420
DatabaseName PubMed
CrossRef
Mechanical & Transportation Engineering Abstracts
Technology Research Database
ProQuest SciTech Collection
ProQuest Technology Collection
Materials Science & Engineering Collection
ProQuest Central (Alumni)
ProQuest Central
ProQuest Central Essentials
ProQuest Central
Technology Collection
ProQuest One Community College
ProQuest Central Korea
Engineering Research Database
SciTech Premium Collection
Civil Engineering Abstracts
ProQuest Engineering Collection
Engineering Database
Publicly Available Content Database
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Academic
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
MEDLINE - Academic
PubMed Central (Full Participant titles)
DOAJ Directory of Open Access Journals
DatabaseTitle PubMed
CrossRef
Publicly Available Content Database
Civil Engineering Abstracts
Engineering Database
Technology Collection
Technology Research Database
Mechanical & Transportation Engineering Abstracts
ProQuest Central Essentials
ProQuest One Academic Eastern Edition
ProQuest Central (Alumni Edition)
SciTech Premium Collection
ProQuest One Community College
ProQuest Technology Collection
ProQuest SciTech Collection
ProQuest Central China
ProQuest Central
ProQuest Engineering Collection
ProQuest One Academic UKI Edition
ProQuest Central Korea
Materials Science & Engineering Collection
Engineering Research Database
ProQuest One Academic
Engineering Collection
MEDLINE - Academic
DatabaseTitleList CrossRef
MEDLINE - Academic

Publicly Available Content Database

PubMed
Database_xml – sequence: 1
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 2
  dbid: NPM
  name: PubMed
  url: https://proxy.k.utb.cz/login?url=http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed
  sourceTypes: Index Database
– sequence: 3
  dbid: 8FG
  name: ProQuest Technology Collection
  url: https://search.proquest.com/technologycollection1
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
EISSN 1099-4300
ExternalDocumentID oai_doaj_org_article_329e5019332c4d65839694aa3dd4e2fb
10_3390_e23111420
34828118
Genre Journal Article
GroupedDBID 29G
2WC
5GY
5VS
8FE
8FG
AADQD
AAFWJ
ABDBF
ABJCF
ACIWK
ADBBV
AEGXH
AENEX
AFKRA
AFZYC
ALMA_UNASSIGNED_HOLDINGS
BCNDV
BENPR
BGLVJ
CCPQU
CS3
DU5
E3Z
ESX
F5P
GROUPED_DOAJ
GX1
HCIFZ
HH5
IAO
ITC
J9A
KQ8
L6V
M7S
MODMG
M~E
NPM
OK1
PGMZT
PIMPY
PROAC
PTHSS
RNS
RPM
TR2
TUS
XSB
~8M
AAYXX
CITATION
7TB
8FD
ABUWG
AZQEC
DWQXO
FR3
KR7
PQEST
PQQKQ
PQUKI
PRINS
7X8
5PM
AFPKN
ID FETCH-LOGICAL-c469t-b035b21f47091eae497faa29d81ec0a16914284bf51c148ef87360531e02c34e3
IEDL.DBID RPM
ISSN 1099-4300
IngestDate Tue Oct 22 15:03:00 EDT 2024
Tue Sep 17 21:19:51 EDT 2024
Sat Oct 26 05:49:03 EDT 2024
Thu Oct 10 15:47:25 EDT 2024
Fri Dec 06 08:59:08 EST 2024
Sat Nov 02 12:30:26 EDT 2024
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 11
Keywords entropy
Cushman method
configuration
Boltzmann
landscape
Language English
License https://creativecommons.org/licenses/by/4.0
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c469t-b035b21f47091eae497faa29d81ec0a16914284bf51c148ef87360531e02c34e3
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
OpenAccessLink https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8617622/
PMID 34828118
PQID 2602033800
PQPubID 2032401
ParticipantIDs doaj_primary_oai_doaj_org_article_329e5019332c4d65839694aa3dd4e2fb
pubmedcentral_primary_oai_pubmedcentral_nih_gov_8617622
proquest_miscellaneous_2604023766
proquest_journals_2602033800
crossref_primary_10_3390_e23111420
pubmed_primary_34828118
PublicationCentury 2000
PublicationDate 20211028
PublicationDateYYYYMMDD 2021-10-28
PublicationDate_xml – month: 10
  year: 2021
  text: 20211028
  day: 28
PublicationDecade 2020
PublicationPlace Switzerland
PublicationPlace_xml – name: Switzerland
– name: Basel
PublicationTitle Entropy (Basel, Switzerland)
PublicationTitleAlternate Entropy (Basel)
PublicationYear 2021
Publisher MDPI AG
MDPI
Publisher_xml – name: MDPI AG
– name: MDPI
References Gao (ref_4) 2018; 22
Gao (ref_10) 2021; 36
Cushman (ref_11) 2015; 30
ref_14
ref_13
ref_12
Cushman (ref_18) 2019; 7
Zenil (ref_21) 2011; 17
Gao (ref_6) 2019; 34
ref_2
ref_19
Shannon (ref_5) 1948; 27
ref_17
ref_16
Liebermann (ref_15) 1972; 11
ref_8
Cushman (ref_1) 2016; 31
Gaucherel (ref_20) 2014; 9
Zhao (ref_9) 2019; 34
ref_7
Gao (ref_3) 2017; 32
References_xml – volume: 34
  start-page: 1849
  year: 2019
  ident: ref_9
  article-title: Calculating spatial configurational entropy of a landscape mosaic based on the Wasserstein metric
  publication-title: Landsc. Ecol.
  doi: 10.1007/s10980-019-00876-x
  contributor:
    fullname: Zhao
– volume: 27
  start-page: 379
  year: 1948
  ident: ref_5
  article-title: A Mathematical Theory of Communication
  publication-title: Bell Syst. Tech. J.
  doi: 10.1002/j.1538-7305.1948.tb01338.x
  contributor:
    fullname: Shannon
– ident: ref_7
  doi: 10.3390/e22090937
– ident: ref_12
– ident: ref_17
  doi: 10.1017/CBO9780511614415
– volume: 9
  start-page: 440
  year: 2014
  ident: ref_20
  article-title: Ecosystem complexity through the lens of logical depth: Capturing ecosystem individuality
  publication-title: Biol. Theory
  doi: 10.1007/s13752-014-0162-2
  contributor:
    fullname: Gaucherel
– volume: 30
  start-page: 7
  year: 2015
  ident: ref_11
  article-title: Thermodynamics in landscape ecology: The importance of integrating measurement and modeling of landscape entropy
  publication-title: Landsc. Ecol.
  doi: 10.1007/s10980-014-0108-x
  contributor:
    fullname: Cushman
– volume: 7
  start-page: 440
  year: 2019
  ident: ref_18
  article-title: Metrics and models for quantifying ecological resilience at landscape scales
  publication-title: Front. Ecol. Evol.
  doi: 10.3389/fevo.2019.00440
  contributor:
    fullname: Cushman
– ident: ref_16
– ident: ref_14
  doi: 10.3390/e23111425
– volume: 36
  start-page: 815
  year: 2021
  ident: ref_10
  article-title: Wasserstein metric-based Boltzmann entropy of a landscape mosaic: A clarification, correction and evaluation of thermodynamic consistency
  publication-title: Landsc. Ecol.
  doi: 10.1007/s10980-020-01177-4
  contributor:
    fullname: Gao
– ident: ref_13
– ident: ref_19
  doi: 10.3390/e23121616
– volume: 11
  start-page: 280
  year: 1972
  ident: ref_15
  article-title: Thermodynamic consistency test methods
  publication-title: Ind. Eng. Chem. Fundam.
  doi: 10.1021/i160042a023
  contributor:
    fullname: Liebermann
– ident: ref_2
  doi: 10.3390/e20040298
– ident: ref_8
  doi: 10.3390/e22040381
– volume: 31
  start-page: 481
  year: 2016
  ident: ref_1
  article-title: Calculating the configurational entropy of a landscape mosaic
  publication-title: Landsc. Ecol.
  doi: 10.1007/s10980-015-0305-2
  contributor:
    fullname: Cushman
– volume: 32
  start-page: 1
  year: 2017
  ident: ref_3
  article-title: A hierarchy-based solution to calculate the configurational entropy of landscape gradients
  publication-title: Landsc. Ecol.
  contributor:
    fullname: Gao
– volume: 22
  start-page: 1046
  year: 2018
  ident: ref_4
  article-title: An efficient analytical method for computing the Boltzmann entropy of a landscape gradient
  publication-title: Trans. GIS
  doi: 10.1111/tgis.12315
  contributor:
    fullname: Gao
– volume: 34
  start-page: 2183
  year: 2019
  ident: ref_6
  article-title: Computation of Boltzmann entropy of a landscape: A review and a generalization
  publication-title: Landsc. Ecol.
  doi: 10.1007/s10980-019-00814-x
  contributor:
    fullname: Gao
– volume: 17
  start-page: 26
  year: 2011
  ident: ref_21
  article-title: Image characterization and classification by physical complexity
  publication-title: Complexity
  doi: 10.1002/cplx.20388
  contributor:
    fullname: Zenil
SSID ssj0023216
Score 2.3511457
Snippet There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the...
SourceID doaj
pubmedcentral
proquest
crossref
pubmed
SourceType Open Website
Open Access Repository
Aggregation Database
Index Database
StartPage 1420
SubjectTerms Boltzmann
configuration
Consistency
Criteria
Cushman method
Entropy
Experiments
landscape
Methods
Normal distribution
Simulation
Standard deviation
SummonAdditionalLinks – databaseName: DOAJ Directory of Open Access Journals
  dbid: DOA
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV07T8MwELYQEwsC8QoUZBBrRGI7iTMCKkIImKjULXL8AIYmVdsM_Hvu7LRqERILWxQ7inOX831fcg9CrgteGAl7P9i3zmLwxzqWjqexU7VygkujNWYjv7zmjyPxNM7Ga62-MCYslAcOgrvhrLQZ4BDOmRYG_CUv81IoxY0Rlrna774JW5KpnmpxluahjhAHUn9jAcVg0miy4X18kf7fkOXPAMk1j_OwR3Z7qEhvwxL3yZZtDkgLep1NWhMayVPfb3OOuPeLto4CmqP33fxjohr64ntD49nQuQF8VBhvG_f53s36r4B0iLHqU3-5os-Y-YsxUXC0wLi4QzJ6GL7dP8Z904RYA9NdxHXCs5qlThSABKyyoiycUqw0MrU6UVgbBxiHqF2WaqBC1smC52iJNmGaC8uPyHbTNvaEUMA2OjUq4ayuBWacGkALmcTsU8u0MRG5WgqzmobaGBVwCpR4tZJ4RO5QzKsJWM7anwAlV72Sq7-UHJHBUklVb2PzCpgYS4BhJ3CPy9UwWAf-8lCNbTs_BwgybKJ5RI6DTlcrwbI-EvhVRIoNbW8sdXOk-fzwFbgl4L6csdP_eLYzssMwTgb8IZMDsr2YdfYcgM6ivvDv9DcvU_wW
  priority: 102
  providerName: Directory of Open Access Journals
– databaseName: ProQuest Technology Collection
  dbid: 8FG
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1LT9wwEB61y6WXCgRtU5bKoF4jEttJnBMCxIIQ9FQkbpHjxy4HkmUfh_57ZpxsyqKqtyh2FMvj8XyfPQ-An4UorMK9H_XbZDHaYxMrL9LY61p7KZQ1hqKR73_lNw_y9jF77A_clr1b5WZPDBu1bQ2dkZ8i7uYJ8qkkOZu_xFQ1im5X-xIaH2En5UVOLn1qcj0QLsHTvMsmJJDanzrEMhQ6mmzZoJCq_1_48r2b5Bu7M9mFzz1gZOedhPfgg2v2oUXpLp5b25WTZ6Hq5pLQ7x_WeoaYjl2ul7Nn3bD7UCGa3nb1G9BSde1t45-m60V_FsiuyGN9Hj7X7I7if8kzCp9W5B13AA-Tq9-XN3FfOiE2yHdXcZ2IrOaplwXiAaedLAuvNS-tSp1JNGXIQd4ha5-lBgmR86oQOemjS7gR0okvMGraxn0DhgjHpFYngte1pLhTi5ghUxSD6rixNoKTzWRW8y5DRoXMgma8GmY8ggua5qEDJbUOL9rFtOp1pBK8dBlCTiG4kRahkSjzUmotrMVf-TqC8UZIVa9py-rvuojgeGhGHaGLD924dh36IE3GrTSP4Gsn02EklNxHIcuKoNiS9tZQt1uap1nIw60Q_eWcf___sA7hEyc_GLR3XI1htFqs3RECmVX9I6zWVwj89BU
  priority: 102
  providerName: ProQuest
Title Thermodynamic Consistency of the Cushman Method of Computing the Configurational Entropy of a Landscape Lattice
URI https://www.ncbi.nlm.nih.gov/pubmed/34828118
https://www.proquest.com/docview/2602033800
https://search.proquest.com/docview/2604023766
https://pubmed.ncbi.nlm.nih.gov/PMC8617622
https://doaj.org/article/329e5019332c4d65839694aa3dd4e2fb
Volume 23
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1La9wwEB6y6aWX0tCX22RRSq_O2pIf8jFZdhNKNoTSwN6MrEeykLWXfRz67zMj20u29NSLMZaEhEbSfJ88D4AfuciNxLMf97dOQ9THOpROxKFTlXKJkEZr8kae3WU3D8nPeTo_grT3hfFG-7paXNTPy4t68eRtK1dLPertxEb3s7FEtZtxPhrAANVvT9E7liV4nLUhhATy-ZFFAEP-opTyjQK5yJjye7zSQT5U_7_w5d9mkq_0zvQ9vOsAI7tsB3YCR7b-AA1Kd71sTJtOnvmsmxtCv39Y4xhiOjbebZ6WqmYznyGavrb5G1BTteVN7RaPu3V3F8gmZLG-8s0VuyX_X7KMwrctWcd9hIfp5Pf4JuxSJ4Qa-e42rCKRVjx2SY54wCqbFLlTihdGxlZHiiLkIO9IKpfGGgmRdTIXGe1HG3EtEis-wXHd1PYLMEQ4OjYqEryqEvI7NYgZUkk-qJZrYwL43k9muWojZJTILGjyy_3kB3BF07yvQEGt_Ydm_Vh2oi0FL2yKkFMIrhOD0EgUWZEoJYzBrlwVwGkvpLLbaZsS-RiPkGdH2Mf5vhj3CP34ULVtdr4O0mQ8SrMAPrcy3Y-kXxMB5AfSPhjqYQkuSx-Hu1uGX_-75Td4y8lEBlUhl6dwvF3v7BlinG01hIGcXg_hzdXk7v7X0N8U4PN6Hg_9an8Bj6wBqg
link.rule.ids 230,314,727,780,784,864,885,2102,12765,21388,27924,27925,33373,33374,33744,33745,43600,43805,53791,53793,74035,74302
linkProvider National Library of Medicine
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1LT9wwEB5ROLQXVFRoU2jrIq4Rie0kzqlqEdst7HICiVvk-AEcSLb7OPDvmXGyabequEVxolgzGc_32fMAOClEYRWu_WjfJovRH5tYeZHGXtfaS6GsMZSNPL3Kxzfy4ja77TfcFn1Y5XpNDAu1bQ3tkZ8i7uYJ8qkk-Tb7HVPXKDpd7VtovIIdKdB1U6b46OdAuARP866akEBqf-oQy1DqaLLhg0Kp_v_hy3_DJP_yO6O3sNsDRva90_AebLnmHbSo3flja7t28ix03VwQ-n1irWeI6djZanH_qBs2DR2i6W7XvwE9VTfeNv7hbjXv9wLZOUWsz8Lrmk0o_5cio_BqSdFx-3AzOr8-G8d964TYIN9dxnUispqnXhaIB5x2siy81ry0KnUm0VQhB3mHrH2WGiREzqsC5Yf26BJuhHTiALabtnEfgCHCManVieB1LSnv1CJmyBTloDpurI3geC3MatZVyKiQWZDEq0HiEfwgMQ8PUFHrcKOd31W9jVSCly5DyCkEN9IiNBJlXkqthbX4KV9HcLRWUtVb2qL6819E8HUYRhuhgw_duHYVnkGajEtpHsH7TqfDTKi4j0KWFUGxoe2NqW6ONA_3oQ63QvSXc_7x5Wl9gdfj6-mkmvy6ujyEN5xiYtD3cXUE28v5yn1CULOsP4c_9xmGgvb3
linkToPdf http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1Lb9QwEB7BVkJcEIhXoIBBXKN1bCdxToiWXRVoVxWiUm-R40fbQ5NlHwf-PTOJN7AIcYtiR7E8Hs_32fMAeF_K0mnc-1G_bZ6iPbapDjJLg2lMUFI7ayka-WxRnFyoL5f5ZfR_Wke3yt2e2G_UrrN0Rj5F3C048inOpyG6RZx_mn9Y_kipghTdtMZyGnfhAK0iFxM4OJotzr-N9EuKrBhyC0kk-lOPyIYCSfmeReoT9_8Lbf7tNPmHFZo_hAcRPrKPg7wfwR3fPoYOZb267dxQXJ71NTjXhIV_si4wRHjseLu-vjUtO-vrRdPboZoD2q2hvWvDzdV2FU8G2Yz815f954adUjQw-Unh04Z85Z7AxXz2_fgkjYUUUovsd5M2XOaNyIIqER1441VVBmNE5XTmLTeULwdZiGpCnlmkRz7oUhaknZ4LK5WXT2HSdq1_Dgzxjs2c4VI0jaIoVIcIItcUkeqFdS6Bd7vJrJdDvowaeQbNeD3OeAJHNM1jB0px3b_oVld11JhaisrnCEClFFY5BEqyKipljHQOfxWaBA53Qqqj3q3r36skgbdjM2oMXYOY1nfbvg-SZtxYiwSeDTIdR0KpfjRyrgTKPWnvDXW_pb257rNya8SChRAv_j-sN3APl219-nnx9SXcF-Qgg4ZQ6EOYbFZb_woRzqZ5HZfuLy5g_JM
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Thermodynamic+Consistency+of+the+Cushman+Method+of+Computing+the+Configurational+Entropy+of+a+Landscape+Lattice&rft.jtitle=Entropy+%28Basel%2C+Switzerland%29&rft.au=Cushman%2C+Samuel+A&rft.date=2021-10-28&rft.eissn=1099-4300&rft.volume=23&rft.issue=11&rft_id=info:doi/10.3390%2Fe23111420&rft_id=info%3Apmid%2F34828118&rft.externalDocID=34828118
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1099-4300&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1099-4300&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1099-4300&client=summon