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  <doc>
    <id>413</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1999-08-09</completedDate>
    <publishedDate>1999-08-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On Conformational Dynamics induced by Langevin Processes</title>
    <abstract language="eng">The function of many important biomolecules is related to their dynamic properties and their ability to switch between different {\em conformations}, which are understood as {\em almost invariant} or {\em metastable} subsets of the positional state space of the system. Recently, the present authors and their coworkers presented a novel algorithmic scheme for the direct numerical determination of such metastable subsets and the transition probability between them. Although being different in most aspects, this method exploits the same basic idea as {\sc Dellnitz} and {\sc Junge} in their approach to almost invariance in discrete dynamical systems: the almost invariant sets are computed via certain eigenvectors of the Markov operators associated with the dynamical behavior. In the present article we analyze the application of this approach to (high--friction) Langevin models describing the dynamical behavior of molecular systems coupled to a heat bath. We will see that this can be related to theoretical results for (symmetric) semigroups of Markov operators going back to {\sc Davies}. We concentrate on a comparison of our approach in respect to random perturbations of dynamical systems.</abstract>
    <identifier type="serial">SC-99-25</identifier>
    <identifier type="opus3-id">414</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-4130</identifier>
    <enrichment key="SourceTitle">Appeared in: Int. Conf. on Differential Equations (EquaDiff 99) B. Fiedler et al. (eds.) 2000, vol. 2,m pp 1247-1262</enrichment>
    <author>Christof Schütte</author>
    <author>Wilhelm Huisinga</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-99-25</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Smoluchowski equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Fokker--Planck equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>semigroup of Markov operators</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>canonical ensemble</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>small noise</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>first exit time</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>half time perio</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="15A18">Eigenvalues, singular values, and eigenvectors</collection>
    <collection role="msc" number="60J25">Continuous-time Markov processes on general state spaces</collection>
    <collection role="msc" number="60J60">Diffusion processes [See also 58J65]</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/413/SC-99-25.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/413/SC-99-25.pdf</file>
  </doc>
  <doc>
    <id>424</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
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    <type>reportzib</type>
    <publisherName/>
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    <creatingCorporation/>
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    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1999-10-18</completedDate>
    <publishedDate>1999-10-18</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems</title>
    <abstract language="eng">The article surveys the development of novel mathematical concepts and algorithmic approaches based thereon in view of their possible applicability to biomolecular design. Both a first deterministic approach, based on the Frobenius-Perron operator corresponding to the flow of the Hamiltonian dynamics, and later stochastic approaches, based on a spatial Markov operator or on Langevin dynamics, can be subsumed under the unified mathematical roof of the transfer operator approach to effective dynamics of molecular systems. The key idea of constructing specific transfer operators especially taylored for the purpose of conformational dynamics appears as the red line throughout the paper. Different steps of the algorithm are exemplified by a trinucleotide molecular system as a small representative of possible RNA drug molecules.</abstract>
    <identifier type="serial">SC-99-36</identifier>
    <identifier type="opus3-id">425</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-4247</identifier>
    <enrichment key="SourceTitle">Appeared in: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. B. Fiedler (ed.) Springer 2001, pp. 191-223</enrichment>
    <author>Christof Schütte</author>
    <author>Wilhelm Huisinga</author>
    <author>Peter Deuflhard</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-99-36</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Transfer operator</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Markov process</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Markov chain</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>molecular dynamics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>biomolecular conformations</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>canonical ensemble</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>transition prohability</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Ha</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="47B15">Hermitian and normal operators (spectral measures, functional calculus, etc.)</collection>
    <collection role="msc" number="60J25">Continuous-time Markov processes on general state spaces</collection>
    <collection role="msc" number="60J60">Diffusion processes [See also 58J65]</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="deuflhard">Deuflhard, Peter</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/424/SC-99-36.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/424/SC-99-36.pdf</file>
  </doc>
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