Abstract

We report on shot-noise limited measurements of the instantaneous velocity distribution of a Brownian particle. Our system consists of a single micron-sized glass sphere held in an optical tweezer in a liquid in equilibrium at room temperature. We provide a direct verification of a modified Maxwell-Boltzmann velocity distribution and modified energy equipartition theorem that account for the kinetic energy of the liquid displaced by the particle. Our measurements confirm the distribution over a dynamic range of more than six orders of magnitude in count-rate and five standard deviations in velocity.

© 2015 Optical Society of America

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References

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  1. K. Huang, Statistical mechanics (Wiley1987).
  2. Lord Kelvin, “On a decisive test-case disproving the Maxwell-Boltzmann doctrine regarding distribution of kinetic energy,” Proc. R. Soc. London 51, 397–399 (1892).
    [Crossref]
  3. R. J. Gould and R. K. Thakur, “Deviation from a Maxwellian velocity distribution in low-density plasmas,” Phys. Fluids,  14, 1701–1706 (1971).
    [Crossref]
  4. R. J. Gould and M. Levy, “Deviation from a Maxwellian velocity distribution in regions of interstellar molecular hydrogen,” Astrophys. J. 206, 435–439 (1976).
    [Crossref]
  5. D. D. Clayton, “Maxwellian relative energies and solar neutrinos,” Nature 249, 131 (1974).
    [Crossref]
  6. T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
    [Crossref] [PubMed]
  7. S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
    [Crossref] [PubMed]
  8. R. Zwanzig and M. Bixon, “Compressibility effects in the hydrodynamic theory of Brownian motion,” J. Fluid Mech. 69, 21–25 (1975).
    [Crossref]
  9. A. B. Basset, “On the motion of a sphere in a viscous liquid,” Phys. Eng. Sci. 179, 43–63 (1888).
    [Crossref]
  10. H. J. H. Clercx and P. P. J. M. Schram, “Brownian particles in shear flow and harmonic potentials: A study of long-time tails,” Phys. Rev. A 46, 1942–1950 (1992).
  11. B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
    [Crossref]
  12. A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
    [Crossref] [PubMed]
  13. I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
    [Crossref] [PubMed]
  14. D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
    [Crossref] [PubMed]
  15. M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
    [Crossref]

2014 (1)

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

2011 (1)

M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
[Crossref]

2010 (2)

D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

2008 (1)

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

2007 (1)

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

1992 (1)

H. J. H. Clercx and P. P. J. M. Schram, “Brownian particles in shear flow and harmonic potentials: A study of long-time tails,” Phys. Rev. A 46, 1942–1950 (1992).

1980 (1)

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[Crossref] [PubMed]

1976 (1)

R. J. Gould and M. Levy, “Deviation from a Maxwellian velocity distribution in regions of interstellar molecular hydrogen,” Astrophys. J. 206, 435–439 (1976).
[Crossref]

1975 (1)

R. Zwanzig and M. Bixon, “Compressibility effects in the hydrodynamic theory of Brownian motion,” J. Fluid Mech. 69, 21–25 (1975).
[Crossref]

1974 (1)

D. D. Clayton, “Maxwellian relative energies and solar neutrinos,” Nature 249, 131 (1974).
[Crossref]

1971 (1)

R. J. Gould and R. K. Thakur, “Deviation from a Maxwellian velocity distribution in low-density plasmas,” Phys. Fluids,  14, 1701–1706 (1971).
[Crossref]

1892 (1)

Lord Kelvin, “On a decisive test-case disproving the Maxwell-Boltzmann doctrine regarding distribution of kinetic energy,” Proc. R. Soc. London 51, 397–399 (1892).
[Crossref]

1888 (1)

A. B. Basset, “On the motion of a sphere in a viscous liquid,” Phys. Eng. Sci. 179, 43–63 (1888).
[Crossref]

Ashkin, A.

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[Crossref] [PubMed]

Basset, A. B.

A. B. Basset, “On the motion of a sphere in a viscous liquid,” Phys. Eng. Sci. 179, 43–63 (1888).
[Crossref]

Bixon, M.

R. Zwanzig and M. Bixon, “Compressibility effects in the hydrodynamic theory of Brownian motion,” J. Fluid Mech. 69, 21–25 (1975).
[Crossref]

Chavez, I.

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

Cheng, D.

D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
[Crossref] [PubMed]

Clayton, D. D.

D. D. Clayton, “Maxwellian relative energies and solar neutrinos,” Nature 249, 131 (1974).
[Crossref]

Clercx, H. J. H.

H. J. H. Clercx and P. P. J. M. Schram, “Brownian particles in shear flow and harmonic potentials: A study of long-time tails,” Phys. Rev. A 46, 1942–1950 (1992).

Florin, E. L.

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Florin, E.-L.

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

Forró, L.

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Franosch, T.

M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
[Crossref]

Gould, R. J.

R. J. Gould and M. Levy, “Deviation from a Maxwellian velocity distribution in regions of interstellar molecular hydrogen,” Astrophys. J. 206, 435–439 (1976).
[Crossref]

R. J. Gould and R. K. Thakur, “Deviation from a Maxwellian velocity distribution in low-density plasmas,” Phys. Fluids,  14, 1701–1706 (1971).
[Crossref]

Grimm, M.

M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
[Crossref]

Halvorsen, K.

D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
[Crossref] [PubMed]

Henderson, K.

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

Huang, K.

K. Huang, Statistical mechanics (Wiley1987).

Huang, R.

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

Jeney, S.

M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
[Crossref]

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Kelvin, Lord

Lord Kelvin, “On a decisive test-case disproving the Maxwell-Boltzmann doctrine regarding distribution of kinetic energy,” Proc. R. Soc. London 51, 397–399 (1892).
[Crossref]

Kheifets, S.

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Kulik, A. J.

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Levy, M.

R. J. Gould and M. Levy, “Deviation from a Maxwellian velocity distribution in regions of interstellar molecular hydrogen,” Astrophys. J. 206, 435–439 (1976).
[Crossref]

Li, T.

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Lukic, B.

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Medellin, D.

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

Melin, K.

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

Raizen, M. G.

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

Schram, P. P. J. M.

H. J. H. Clercx and P. P. J. M. Schram, “Brownian particles in shear flow and harmonic potentials: A study of long-time tails,” Phys. Rev. A 46, 1942–1950 (1992).

Simha, A.

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

Sviben, Ž.

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Thakur, R. K.

R. J. Gould and R. K. Thakur, “Deviation from a Maxwellian velocity distribution in low-density plasmas,” Phys. Fluids,  14, 1701–1706 (1971).
[Crossref]

Wong, W. P.

D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
[Crossref] [PubMed]

Zwanzig, R.

R. Zwanzig and M. Bixon, “Compressibility effects in the hydrodynamic theory of Brownian motion,” J. Fluid Mech. 69, 21–25 (1975).
[Crossref]

Astrophys. J. (1)

R. J. Gould and M. Levy, “Deviation from a Maxwellian velocity distribution in regions of interstellar molecular hydrogen,” Astrophys. J. 206, 435–439 (1976).
[Crossref]

J. Fluid Mech. (1)

R. Zwanzig and M. Bixon, “Compressibility effects in the hydrodynamic theory of Brownian motion,” J. Fluid Mech. 69, 21–25 (1975).
[Crossref]

Nature (1)

D. D. Clayton, “Maxwellian relative energies and solar neutrinos,” Nature 249, 131 (1974).
[Crossref]

Phys. Eng. Sci. (1)

A. B. Basset, “On the motion of a sphere in a viscous liquid,” Phys. Eng. Sci. 179, 43–63 (1888).
[Crossref]

Phys. Fluids (1)

R. J. Gould and R. K. Thakur, “Deviation from a Maxwellian velocity distribution in low-density plasmas,” Phys. Fluids,  14, 1701–1706 (1971).
[Crossref]

Phys. Rev. A (1)

H. J. H. Clercx and P. P. J. M. Schram, “Brownian particles in shear flow and harmonic potentials: A study of long-time tails,” Phys. Rev. A 46, 1942–1950 (1992).

Phys. Rev. E (1)

B. Lukić, S. Jeney, Ž. Sviben, A. J. Kulik, E. L. Florin, and L. Forró, “Motion of a colloidal particle in an optical trap,” Phys. Rev. E 76, 011112 (2007).
[Crossref]

Proc. R. Soc. London (1)

Lord Kelvin, “On a decisive test-case disproving the Maxwell-Boltzmann doctrine regarding distribution of kinetic energy,” Proc. R. Soc. London 51, 397–399 (1892).
[Crossref]

Rev. Sci. Instrum. (2)

I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008).
[Crossref] [PubMed]

D. Cheng, K. Halvorsen, and W. P. Wong, “Note: High-precision microsphere sorting using velocity sedimentation,” Rev. Sci. Instrum. 81, 026106 (2010).
[Crossref] [PubMed]

Science (3)

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[Crossref] [PubMed]

T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science 328, 1673–1675 (2010).
[Crossref] [PubMed]

S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref] [PubMed]

Soft Matter (1)

M. Grimm, S. Jeney, and T. Franosch, “Brownian motion in a Maxwell fluid,” Soft Matter 7, 2076–2084 (2011).
[Crossref]

Other (1)

K. Huang, Statistical mechanics (Wiley1987).

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Figures (5)

Fig. 1
Fig. 1 A simplified schematic of experimental setup for measuring instantaneous velocity of a microsphere trapped by counter-propagating 1064 nm and 532 nm laser beams focused by microscope objectives (OBJ) in liquid. The 1064 nm laser is used to detect the horizontal motion of the particle using a high-power, high bandwidth balanced detector. DM: dichroic mirror, CM: D-shaped mirror.
Fig. 2
Fig. 2 Scanning electron microscope images of the microspheres (sputtered with about 10 nm Au/Pd with 60/40 ratio) demonstrate high sphericity. A: The widely-dispersed (0.1–10 μm) BaTiO3 glass microspheres; B: The mono-disperse silica glass microspheres.
Fig. 3
Fig. 3 (A): The MSD of a typical 1-s trajectory for a trapped microsphere in liquid. Red dashed lines indicate the MSD of a particle moving at constant velocity v r m s * = k B T / m *; black lines are theoretical MSD. (B): The PPSD for the same trajectories. The PPSD flattens at high frequency due to shot noise of the detection beam at 2.4 fm / H z, 9.1 fm / H z and 1.8 fm / H z for a silica microsphere in water, a silica microsphere in acetone and a BaTiO3 microsphere in acetone respectively. The black line is the sum of theoretical PPSD and a constant shot noise. For both plots: magenta diamonds represent silica in water data; green squares represent silica in acetone data; blue circles represent BaTiO3 in acetone data.
Fig. 4
Fig. 4 The VPSD (top) and normalized cumulative VPSD (bottom, normalized to kBT/m*) of a microsphere in liquid for the same trajectories as in Fig. 3 in three systems, A: a silica microsphere in water; B: a silica microsphere in acetone; C: a BaTiO3 microsphere in acetone. In all plots: green squares represent the measurements with trapped particles; blue diamonds represent the noise measured without particles present but with the same detection power; red circles represent the net measurement with noise subtracted; black lines are the theoretical prediction; magenta lines are the shot noise using the results shown in Fig. 3(b), indicating that it is the dominant noise source.
Fig. 5
Fig. 5 The normalized velocity distribution for three systems A: a silica microsphere in water ( v r m s * = 327 μ m / s ); B: a silica microsphere in acetone ( v r m s * = 227 μ m / s ); C: a BaTiO3 microsphere in acetone ( v r m s * = 104 μ m / s ), calculated from 3.6 billion, 200 million and 144 million data points respectively. The histogram bin size for each velocity distribution was set to the rms magnitude of the corresponding noise. For all plots: red circles represent the measurements with trapped microspheres; green diamonds represent the measurements acquired without particles present, but with matching detection power; black lines are the modified MBD predictions; blue lines (overlapping with the black line in C) are Gaussian fits of the measurements, from which the fraction of the mean kinetic energy observed was determined.

Tables (1)

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Table 1 The summary of the results for the three systems.

Equations (1)

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m * x ¨ ( t ) = K x ( t ) 6 π η r x ˙ ( t ) 6 r 2 π ρ f η 0 t ( t t ) 1 / 2 x ¨ ( t ) d t + F i h ( t )

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