Abstract

Understanding the formation of electrodynamically interacting assemblies of metal nanoparticles requires accurate computational methods for determining the forces and propagating trajectories. However, since computation of electromagnetic forces occurs on attosecond to femtosecond timescales, simulating the motion of colloidal nanoparticles on milliseconds to seconds timescales is a challenging multi-scale computational problem. Here, we present a computational technique for performing accurate simulations of laser-illuminated metal nanoparticles. In the simulation, we self-consistently combine the finite-difference time-domain method for electrodynamics (ED) with Langevin dynamics (LD) for the particle motions. We demonstrate the ED-LD method by calculating the 3D trajectories of a single 100-nm-diameter Ag nanoparticle and optical trapping and optical binding of two and three 150-nm-diameter Ag nanoparticles in simulated optical tweezers. We show that surface charge on the colloidal metal nanoparticles plays an important role in their optically driven self-organization. In fact, these simulations provide a more complete understanding of the assembly of different structures of two and three Ag nanoparticles that have been observed experimentally, demonstrating that the ED-LD method will be a very useful tool for understanding the self-organization of optical matter.

© 2015 Optical Society of America

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References

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2015 (1)

Z. Yan, M. Sajjan, and N. F. Scherer, “Fabrication of a material assembly of silver nanoparticles using the phase gradients of optical tweezers,” Phys. Rev. Lett. 114, 143901 (2015).
[Crossref] [PubMed]

2014 (3)

Z. Yan, S. Gray, and N. F. Scherer, “Potential energy surfaces and reaction pathways for light-mediated self-organization of metal nanoparticle clusters,” Nat. Commun. 5, 3751 (2014).
[Crossref] [PubMed]

T. M. Grzegorczyk, J. Rohner, and J.-M. Fournier, “Optical mirror from laser-trapped mesoscopic particles,” Phys. Rev. Lett. 112, 023902 (2014).
[Crossref] [PubMed]

Y. Bao, Z. Yan, and N. F. Scherer, “Optical printing of electrodynamically coupled metallic nanoparticle arrays,” J. Phys. Chem. C 118, 19315–19321 (2014).
[Crossref]

2013 (7)

B. Leimkuhler and C. Matthews, “Robust and efficient configurational molecular sampling via Langevin dynamics,” J. Chem. Phys. 138, 174102 (2013).
[Crossref] [PubMed]

B. Leimkuhler and C. Matthews, “Rational construction of stochastic numerical methods for molecular sampling,” Appl. Math. Res. Express 2013, 34–56 (2013).

O. M. Marago, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat Nano 8, 807–819 (2013).
[Crossref]

G. Volpe and G. Volpe, “Simulation of a brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

Z. Yan, R. A. Shah, G. Chado, S. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS Nano 7, 1790–1802 (2013).
[Crossref] [PubMed]

I. Capoglu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the fdtd method,” Opt. Express 21, 87–101 (2013).
[Crossref] [PubMed]

2012 (3)

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

P. Wang, B. Huang, Y. Dai, and M.-H. Whangbo, “Plasmonic photocatalysts: Harvesting visible light with noble metal nanoparticles,” Phys. Chem. Chem. Phys. 14, 9813–9825 (2012).
[Crossref] [PubMed]

Z. Yan, J. E. Jureller, J. Sweet, M. J. Guffey, M. Pelton, and N. F. Scherer, “Three-dimensional optical trapping and manipulation of single silver nanowires,” Nano Lett. 12, 5155–5161 (2012).
[Crossref] [PubMed]

2010 (7)

G. Baffou, R. Quidant, and F. J. G. de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4, 709–716 (2010). PMID: .
[Crossref] [PubMed]

A. S. Urban, A. A. Lutich, F. D. Stefani, and J. Feldmann, “Laser printing single gold nanoparticles,” Nano Lett. 10, 4794–4798 (2010).
[Crossref] [PubMed]

M. J. Guffey and N. F. Scherer, “All-optical patterning of au nanoparticles on surfaces using optical traps,” Nano Lett. 10, 4302–4308 (2010).
[Crossref] [PubMed]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light–matter interactions with nanoplasmonics,” Small 6, 2498–2507 (2010).
[Crossref] [PubMed]

T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B: At. Mol. Opt. Phys. 43, 102001 (2010).
[Crossref]

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[Crossref]

2009 (1)

R. A. Nome, M. J. Guffey, N. F. Scherer, and S. K. Gray, “Plasmonic interactions and optical forces between au bipyramidal nanoparticle dimers,” J. Phys. Chem. A 113, 4408–4415 (2009).
[Crossref] [PubMed]

2008 (5)

Z. Li, M. Käll, and H. Xu, “Optical forces on interacting plasmonic nanoparticles in a focused gaussian beam,” Phys. Rev. B 77, 085412 (2008).
[Crossref]

R. Gordon, M. Kawano, J. T. Blakely, and D. Sinton, “Optohydrodynamic theory of particles in a dual-beam optical trap,” Phys. Rev. B 77, 245125 (2008).
[Crossref]

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[Crossref] [PubMed]

2007 (3)

2006 (1)

2005 (2)

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

2004 (1)

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4, 899–903 (2004).
[Crossref]

2003 (1)

S. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68, 045415 (2003).
[Crossref]

1999 (1)

T. Jensen, L. Kelly, A. Lazarides, and G. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Clust. Sci. 10, 295–317 (1999).
[Crossref]

1990 (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical matter: Crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref] [PubMed]

1986 (1)

R. Biswas and D. R. Hamann, “Simulated annealing of silicon atom clusters in langevin molecular dynamics,” Phys. Rev. B 34, 895–901 (1986).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1958 (1)

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Statist. 29, 610–611 (1958).
[Crossref]

Allen, P.

P. Allen, D. Frenkel, and J. Talbot, Molecular Dynamics Simulation Using Hard Particles (North-Holland, 1989).

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref] [PubMed]

Andrews, D. L.

T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B: At. Mol. Opt. Phys. 43, 102001 (2010).
[Crossref]

Backman, V.

Baffou, G.

G. Baffou, R. Quidant, and F. J. G. de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4, 709–716 (2010). PMID: .
[Crossref] [PubMed]

Bain, C. D.

Bao, Y.

Y. Bao, Z. Yan, and N. F. Scherer, “Optical printing of electrodynamically coupled metallic nanoparticle arrays,” J. Phys. Chem. C 118, 19315–19321 (2014).
[Crossref]

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Biswas, R.

R. Biswas and D. R. Hamann, “Simulated annealing of silicon atom clusters in langevin molecular dynamics,” Phys. Rev. B 34, 895–901 (1986).
[Crossref]

Blakely, J. T.

R. Gordon, M. Kawano, J. T. Blakely, and D. Sinton, “Optohydrodynamic theory of particles in a dual-beam optical trap,” Phys. Rev. B 77, 245125 (2008).
[Crossref]

Borghese, F.

Box, G. E. P.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Statist. 29, 610–611 (1958).
[Crossref]

Brzobohatý, O.

V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[Crossref] [PubMed]

Burns, M. M.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical matter: Crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref] [PubMed]

Butt, H.-J.

H.-J. Butt, K. Graf, and M. Kappl, Physics and Chemistry of Interfaces (Wiley-VCH, 2003).
[Crossref]

Capoglu, I.

Chado, G.

Z. Yan, R. A. Shah, G. Chado, S. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS Nano 7, 1790–1802 (2013).
[Crossref] [PubMed]

Chan, C. T.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Chen, J.

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cižmár, T.

T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B: At. Mol. Opt. Phys. 43, 102001 (2010).
[Crossref]

V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[Crossref] [PubMed]

Dai, Y.

P. Wang, B. Huang, Y. Dai, and M.-H. Whangbo, “Plasmonic photocatalysts: Harvesting visible light with noble metal nanoparticles,” Phys. Chem. Chem. Phys. 14, 9813–9825 (2012).
[Crossref] [PubMed]

de Abajo, F. J. G.

G. Baffou, R. Quidant, and F. J. G. de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4, 709–716 (2010). PMID: .
[Crossref] [PubMed]

Demergis, V.

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

Denti, P.

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[Crossref]

T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B: At. Mol. Opt. Phys. 43, 102001 (2010).
[Crossref]

V. Karásek, T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. 101, 143601 (2008).
[Crossref] [PubMed]

G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, “Transverse particle dynamics in a bessel beam,” Opt. Express 15, 13972–13987 (2007).
[Crossref] [PubMed]

Feldmann, J.

A. S. Urban, A. A. Lutich, F. D. Stefani, and J. Feldmann, “Laser printing single gold nanoparticles,” Nano Lett. 10, 4794–4798 (2010).
[Crossref] [PubMed]

Fennerty, T. A.

Fernández-Domínguez, A. I.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light–matter interactions with nanoplasmonics,” Small 6, 2498–2507 (2010).
[Crossref] [PubMed]

Fernández-García, R.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light–matter interactions with nanoplasmonics,” Small 6, 2498–2507 (2010).
[Crossref] [PubMed]

Ferrari, A. C.

O. M. Marago, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat Nano 8, 807–819 (2013).
[Crossref]

Florin, E.-L.

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

Fournier, J.-M.

T. M. Grzegorczyk, J. Rohner, and J.-M. Fournier, “Optical mirror from laser-trapped mesoscopic particles,” Phys. Rev. Lett. 112, 023902 (2014).
[Crossref] [PubMed]

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical matter: Crystallization and binding in intense optical fields,” Science 249, 749–754 (1990).
[Crossref] [PubMed]

Frenkel, D.

P. Allen, D. Frenkel, and J. Talbot, Molecular Dynamics Simulation Using Hard Particles (North-Holland, 1989).

Garcés-Chávez, V.

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ACS Nano (2)

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Z. Yan, R. A. Shah, G. Chado, S. Gray, M. Pelton, and N. F. Scherer, “Guiding spatial arrangements of silver nanoparticles by optical binding interactions in shaped light fields,” ACS Nano 7, 1790–1802 (2013).
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Am. J. Phys. (1)

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Electrophoresis (1)

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J. Chem. Phys. (1)

B. Leimkuhler and C. Matthews, “Robust and efficient configurational molecular sampling via Langevin dynamics,” J. Chem. Phys. 138, 174102 (2013).
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J. Clust. Sci. (1)

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Supplementary Material (2)

NameDescription
» Visualization 1: MOV (3889 KB)      Movie showing the trajectories of a 100-nm-diameter Ag nanoparticle that is illuminated with a focused Gaussian beam and one that has no incident source beam
» Visualization 2: MOV (3873 KB)      Movie showing the trajectories of a 150-nm-diameter Ag nanoparticle that is illuminated with a line trap and one that has no incident source beam

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Figures (9)

Fig. 1
Fig. 1 Illustration of the simulation domain and electric field intensity from a focused incident wave. Schematic representing the geometry of the simulation showing (a) the xy and (b) the yz plane. The total-field scattered-field (TF/SF) and convolutional perfectly matched layer (CPML) boundaries are also shown. An example of the electric field intensity (electric field polarized along y) distribution in the yz plane (c) showing the focusing effect of the lens and (d) with a Ag nanoparticle showing the near-field intensity enhancement. The axes in panels (c) and (d) represent grid cells numbers. A grid cell is a cubic box with each side of length 5 nm.
Fig. 2
Fig. 2 Flow chart of the coupled ED-LD method. The outer loop contains the LD update equations. The inner loop containing the FDTD update equations is run every third time step of the outer loop.
Fig. 3
Fig. 3 Results of an ED-LD simulation of a 100-nm-diameter Ag nanoparticle in a focused Gaussian optical beam. (a) Trajectory of the nanoparticle with (maroon trace) and without a source (black trace) (see Visualization 1). The electric field [polarized along y] intensity of the focused Gaussian beam in an xy plane 80 nm from the water-glass interface is shown as the colored background. (b) Optical forces mapped in the small region of the xy plane for the Ag nanoparticle when illuminated with the focused Gaussian beam.
Fig. 4
Fig. 4 Trapping along the z direction at the water-glass interface. In this simulation, the glass surface is located at the z = 600 nm plane. Center-of-mass positions of a 100-nm-diameter Ag particle in a focused Gaussian beam in (a) the xz plane and (b) the yz plane. (c) Potentials of mean force (pmf) in units of kBT as a function of the distance of the center of the nanoparticle from the water-glass interface. The solid curve is a parabolic fit to the pmf.
Fig. 5
Fig. 5 Two-dimensional potentials of mean force determined from histograms of the positions calculated for a 100-nm-diameter Ag nanoparticle illuminated by a focused Gaussian beam for varying focal lengths. The distance between the water-glass interface and focal length of the lens are (a) 202 nm, (b) 107 nm, and (c) 32 nm. Refractive index of the lens (nl) and the corresponding focal lengths (f) are (a) nl = 2.0, f = 1110 nm, (b) nl = 2.062, f = 1015 nm, and (c) nl = 2.121, f = 940 nm. The corresponding peak intensity of the electric field at the water-glass interface is 2.5 MW/cm2, 5 MW/cm2, and 8 MW/cm2, respectively.
Fig. 6
Fig. 6 Optical trapping and binding of two 150-nm-diameter Ag nanoparticles. (a) The trajectories of the two nanoparticles in a line trap along x (maroon and pink traces) and without any incident field (black and grey traces) (see Visualization 2). The electric field (polarized along y) intensity of the line trap in the xy plane at the glass-water interface is shown as the colored background. (b) Histogram (bin width 5 nm) of the separation between the two Ag nanoparticles in the line trap (maroon) and without any incident wave (black) calculated from a simulation of 2 ms total duration.
Fig. 7
Fig. 7 Comparing optical-binding separations of two 150-nm-diameter Ag nanoparticles. (a) Trajectories of the nanoparticles under plane-wave illumination (green and light green traces) and in a line trap (maroon and pink traces) starting from the same pair of initial positions. (b) Histogram of the separation between the two Ag nanoparticles and (c) Potential of mean force calculated as a function of the separation for the different incident waves (Solid lines are parabolic fits). (d) Average separation between two Ag nanoparticles in a line trap (maroon squares) and under plane wave illumination (green circle) as a function of the ζ-potential. Error bars denote standard deviation from the mean.
Fig. 8
Fig. 8 ED-LD simulation of two 150-nm-diameter Ag nanoparticles in a focused Gaussian beam. Trajectories and final assembly of the two nanoparticles (blue and maroon lines) assuming ionic concentrations of (a) 0.02 mmol/L and (b) 0.18 mmol/L. The mean trapped center-of-mass position of (a) 200 nm and (b) 525 nm is shown by the black circles (diameter 150 nm) and initial position (same in both panels) is shown by the X-marks. The direction of polarization of the incident beam is along the y axis. The LD time step used for these simulations was 0.25 μs and the total duration was 1 ms.
Fig. 9
Fig. 9 ED-LD simulation of three 150-nm-diameter Ag nanoparticles in a focused Gaussian optical beam. The X marks represent the initial positions, which are same in all panels. The circle of radius 75 nm, represents the average position of the nanoparticle (averaged over the last 1000 LD time steps). The ζ-potential is fixed at −50 mV, while the ionic concentration (I, in units of mmol/L) is varied to simulate different surface charge conditions: (a) I = 0.01, (b) I = 0.02, (c) I = 0.03, (d) I = 0.04, (e) I = 0.05, (f) I = 0.06, (g) I = 0.07, and (h) I = 0.08. The electric field of the incident beam is polarized along the y axis. Note that the initial positions are identical in all the panels.

Equations (12)

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F = S d s T ¯ n ^ ,
T ¯ = 1 2 [ ( ε ε 0 E c E c * + μ μ 0 H c H c * ) I ¯ 2 ( ε ε 0 | E c | 2 + μ μ 0 | H c | 2 ) ] ,
m d 2 r d t 2 = F ( r , t ) λ v d r d t + η .
m d r d t = p ,
d p d t = F ( r , t ) γ p + m σ ( t ) ,
p n + 1 / 3 = p n + Δ t 2 F ( r n ) ( B )
r n + 1 / 2 = r n + Δ t 2 p n + 1 / 3 m ( A )
p n + 2 / 3 = e γ Δ t p n + 1 / 2 + k B T ( 1 e 2 γ Δ t ) m R n ( O )
r n + 1 = r n + 1 / 2 + Δ t 2 p n + 2 / 3 m ( A )
p n + 1 = p n + 2 / 3 + Δ t 2 F ( r n + 1 ) ( B )
pmf ( ξ ) = k B T ln P ( ξ )
κ = k B T ξ 2

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