brownian motion experiment smoke cell

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1. Brownian motion

Brownian Motion: Evidence for a theory about the nature of gases and liquids

We're constantly surround by air molecules which are bumping into us, moving in random directions. In a liquid, the molecules or atoms are moving around each other, again, randomly and in a solid they're held in position and can only vibrate.

So how do we know that these gas or liquid particles are moving randomly? How do we know that they're particles at all?

Well, one experiment which adds evidence to support this 'kinetic' theory is called 'Brownian Motion'. To set up this experiment, we need:

• a glass 'cell'
• smoke from a glowing splint
• a microscope with a light to illuminate the smoke particles

The smoke is trapped in the glass cell and the microscope is used to observe the motion of the smoke particles.

The tiny smoke particles are observed to move around in a jerky fashion - called a ' random walk '. The random movement of these smoke particles is evidence that, in this case, air molecules are randomly bombarding the smoke particles.

Sometimes more air molecules collide with the left side of a smoke particle than on the right. This means that the smoke particle gets an overall push (a resultant force) to the right, and so moves to the right.

The next moment, there may be more collisions of air molecules on another side of the smoke particle and this would make the smoke particle change direction.

This random bombardment causes the smoke particle's 'random walk', which is evidence of the kinetic theory of matter - that gases and liquids are made up of tiny invisible particles that move randomly and collide with themselves and other objects.

You can also do this experiment with a liquid by sprinkling pollen grains onto water and then viewing the 'random walk' of the pollen grains with a microscope. 'Brownian Motion' is named after the scientist Robert Brown who carried out the experiment in this way.

GCSE Physics Keywords: Gas, Liquid, Particles, Motion, Collisions, Random, Brownian motion

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Notes on demonstrating Brownian motion: The pros and cons of the Whitley Bay smoke cell compared with observing very dilute milk

by David Walker, UK

Update November 2013: HD 1080P extended video clip of Brownian motion in dilute milk available to download. See below.

I first remember being shown Brownian motion in high school by the physics teacher using a microscope to observe smoke particles suspended in air in a glass cell. This is one of a number of recommended ways of demonstrating this classic experiment; others include observing aqueously suspended fine particles such as pigments (e.g. Gamboge, Indian ink) or fat globules in very dilute milk (refs. 1 - 3).  Robert Brown studied the motion after whom it takes its name, in tiny particles released from the medium within pollen grains ( not the pollen grains themselves which are too large to show the motion, ref. 4).

I was browsing eBay recently and noticed a used Brownian motion smoke cell which I purchased for a few pounds as was interested in comparing its effectiveness with other techniques tried; my preferred method as a hobbyist is to use extremely dilute milk as it is effective and fairly easy to set up. (See Micscape ' Home in close-up ' article.)

The smoke cell purchased was the 'Whitley Bay' pattern made by Griffin & George Ltd, which seems a popular design and is still available new from educational suppliers . There has been speculation online as to why it was apparently named after a small town on the north east coast of England (ref. 5). Was the designer based there; or perhaps an allusion of the smoke cell's lighting to the town's offshore lighthouse? I'd be interested to hear from any reader who knows why. Note added March 5th 2014 and updated March 19th 2018: Thank you to a reader Albert and also Kate Le Sueur who both note that it was called the Whitley Bay Smoke Cell because it was designed and invented by the physics department at the Whitley Bay Grammar School now the Whitley Bay High School.

The cell is shown below. It is essentially a self contained lamp, lens and smoke cell to observe the bright smoke particles under a microscope against a dark background. It uses the 'festoon' type bulb* 12V 3W, to shine light through a solid glass rod to give a thin beam of light in the small cell. As smoke particles pass through this beam the particles are seen as white specks undergoing motion, both Brownian and other motion like convection. *The bulbs are readily available as they are used in e.g. some older car door courtesy lamps.

Cell with light shade off for clarity. 1 - festoon bulb, 2 - solid glass rod acts as lens, 3 - small glass cell sits in friction collar in base, with cover slip, 4 - terminals for 12V supply.

Cell with light guard on and smoke cell removed and sitting on this cover to show the cell more clearly. This example has been well used as cell has tarry deposits in base. The instructions note the importance of regularly cleaning the cell walls.

Some practical disadvantages of both the design and its ability to unambiguously demonstrate Brownian motion were discussed by Marcus Rowland, a head science technician at a London school, in a 'New Scientist Forum' article 'The great Brownian motion swindle' (ref. 6), which prompted some letters from readers (ref. 7).

Having experimented with the cell for some time, the view of the smaller smoke articles brightly seen in motion against a black background is quite striking although simpler methods seem better at more clearly demonstrating Brownian motion.

Observations of the smoke cell design and its use:

There a number of resources online that describe the cell's use, e.g. on the ' Practical Physics ' website.

It's suggested that paper drinking straws or corrugated paper is set alight then extinguished to 'pour' smoke into the cell; although the author rolled up some paper round a '00' paintbrush handle and taped in middle, then remove the 'former', and used this. Once the coverslip is put on, the smoke can be studied for some while, 5 - 10 mins at least from the author's trials.

A coverslip can easily slip off the top of the small cell, and the cell in the author's example was also not very tightly held at its base; disadvantages also noted in Rowland's article. A few tiny dabs of eg Vaseline on the cell's top face help prevent the coverslip shifting (not water which tends to cause condensation in the warm lighting.)

Tracing paper in plane of cell showing how the glass rod creates a thin beam of light. Graduations are in mm.

Using tracing paper, the lit area is about 4-5 mm below the top of the cell. This limits the objective used to those that have at least this working distance plus ideally a few more mm for safety. A 10x objective is recommended, and the author's Zeiss 10X NA0.22 has sufficient working distance, but the author had a preference for the Zeiss 6.3x NA0.16, as the wider field showed more particles and gave a safer working distance.

As the online resources recommend, starting with the objective by eye very close to the coverslip then focussing upward until the particles are seen works well.

The lighting is not that strong but the bright smoke particles can readily be seen by eye with typical 10x eyepieces; a slightly darkened room helps. To video the motion the author used a typical domestic C-mount monochrome security camera, sensitivity 0.05 Lux, which was just sensitive enough to record the particles in motion. Strong LED lighting is now available in the festoon bulb format for a few pounds, perhaps these bulbs may be more effective. Such lamps would also not emit infrared so may help minimise convection in the cell.

The random motion of the smaller smoke particles can be seen but the large volume of the cell relative to the particles and tungsten side lighting can introduce motion by convection as well. The narrow depth of field of the objective and tall cell also means that a given particle is only seen for a short time before they drift out of focus.

The advantages of studying very dilute milk:

From the author's past and more recent comparative trials using a drop of very dilute milk on a microscope slide with cover slip, this method does seems to offer a number of advantages to the smoke cell:

Apart from the microscope no special apparatus is needed, just a microscope slide, coverslip and some simple tools (see procedure below, click image).

If the coverslip is temporarily sealed along the edges with Vaseline, the initial currents die down in a few minutes and the fat globule motion by inspection is almost entirely Brownian.

The very thin liquid layer also ensures fat globules remain in focus, and the random motion can be studied more carefully.

The variation in sizes of the fat globules also allows the motion of smaller globules to be compared with that of larger globules.

Using normal transmitted light with the microscope allows much easier visual or video studies to be carried out.

Video comparison - smoke cell cf dilute milk Videos use Xvid compression, Codecs here if doesn't play on Windows / Linux PC.   If videos don't autoplay when click image, use the right mouse button to click image to 'save target' to PC, then view in PC's own player.   Camera: Hunt HTB-686D 0.05 Lux 580 TV line B/W security camera with Motic 16mm relay lens. A near infra red blocking filter was used, and vital, as these sort of B/W cameras often don't have one. Video recorded to domestic DVD recorder. Then file edited in Virtual Dub.

Smoke cell, left 6.3x objective with 5x eyepiece, right 10x objective with 10x eyepiece, (2 Mbyte and 1 Mbyte clips, 30 secs each). In the author's trials, motion by convection varied depending on the focus plane, the above clips used focus planes to mimimise such motion. A 10x objective is suggested in the maker's instructions but the depth of field becomes quite small with smoke particles not being observed for long before moving out of focus. A 6.3x objective was better in this respect; the lefthand clip above shows some motion by convection (along y axis in clip) in addition to Brownian motion. Smoke was generated with coiled paper which gave particles of similar size, thus not allowing a study of the effect on Brownian motion on different sized particles. Fat globules in milk is better to demonstrate this aspect.

Very dilute milk in water: temporary sealed coverslip. Left: brightfield, (2 Mbytes, 27 secs), optic mag 400x, (Zeiss 40x NA0.75 objective, 10x eyepiece.) Right: darkfield, (8 Mbytes, 42 secs), optical mag 320x, (Zeiss 16x NA0.4 objective, 10x eyepiece, 2x Optovar.) Motion by convection is virtually eliminated if the slip is sealed well and allowed to equilibrate for a few minutes; allowing Brownian motion to be more readily studied. The thin liquid layer keeps more fat globules in focus so can be viewed for longer. The difference in motion of larger and smaller fat globules can also be seen. To minimise the liquid thickness, the slip can be sealed by immersion oil instead of the thicker Vaseline. Some experimentation is worthwhile with the milk dilution, too little and the fat globule density confuses the study (and gives glare in darkfield), too dilute and there's not a good range of particles. The concentration of the milk in above clip is a little on the high side for clear darkfield.

To study individual fat globules more closely eg for statistical analysis, inserting the C mount video camera directly into eyepiece with an adaptor introduces a useful extra mag caused by the small sensor crop.

Film clip details, 3 Mbytes, 10 secs, Cinepak Codec compression: Very dilute milk taken with a Russian 40X (N.A. 0.65) achromatic objective in bright field illumination on a LOMO Biolam. The image was projected into the video camera (Panasonic CL350 430 line resolution) without an eyepiece. The droplets range in size from about 0.5 to 3 �m (1 �m is 1/1000th of a millimetre). The milk dilution was about optimum, with a good range of different sizes to study individual particles without an excess cluttering the field. Video clip from an earlier article ' Microscopy around the home '.

In transmitted brightfield, the effect is perhaps not as striking as the bright smoke particles against a dark background in the smoke cell, although if the milk is extremely dilute, darkfield can be used with the milk for a similar effect. Darkfield can be readily setup on many student microscopes likely to be in schools using a suitable patchstop in the condenser filter tray.

A Google search suggests that dilute milk or other aqueous suspended solids are widely used for the demonstration, including statistical studies of particle movement.

Smoke cell cf very dilute milk, sumary of pros and cons. (Personal view based on author's trials as a hobbyist who does not have firsthand experience of their use with students.)

Comments to the author David Walker are welcomed.

  References and resources

1) Google Book Search - early mentions of using gamboge suspended in liquids to study Brownian motion in 'The Treasures of Botany', 1866 and 'The Popular Science Monthly' 1915.

2) Google Book Search - early mention of using Indian ink to demonstrate Brownian motion in ' The Principles of Bacteriology a Practical Manual for Students and Physicians', by A C Abbott, 1915. (To distinguish this motion from that of live bacteria.)

3) Google Book Search - early observation that fat globules in milk undergo Brownian motion, 'Popular Science' magazine, Dec. 1876. vol. 10, no. 9. p139.

4) 'Brownian movement in Clarkia pollen: a reprise of the first observations' by Brian J. Ford. The Microscope , 1992, 40 (4), pp. 235-241. This fascinating paper describes how Robert Brown's observations were repeated using one of Brown's original microscopes. This demonstration was carried out to dispel the suggestion that Brown's microscopes were not good enough to show the motion. The paper also discusses many aspects of Brown's life and work. This paper is on Brian Ford's web site .

5) Whitley Bay , Wikipedia entry with photo of lighthouse.

6) ' The great Brownian motion swindle. Marcus L Rowland uncovers a case of mistaken identity .', New Scientist, Forum, 13 April, 1991, p. 49.

7) ' Browned off ' - readers' letters (nos. 1, 3 and 5 in link list) in later issues of New Scientist in response to above article.

8) ' Brown's Brownian motion revisited ' Science News, August 15, 1992. A recent controversy between scientists as to whether Brown observed the motion.

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Instructional Resources and Lecture Demonstrations

4d10.10 - brownian motion.

Place the microslide with the desired density of microspheres under the microscope.  Using the 100 power lens you may observe the Brownian motion. 

Poisson Distribution.  THIS WILL TAKE 2 TO 3 DAYS ADVANCED NOTICE:   Prepare a microslide with a high density solution of microspheres.  Set the microslide at an incline for several days or until a Poisson distribution is established in the microslide.  

Put a drop of microspheres ( 1 micron diameter spheres ) onto the microscope slide.  Cover with a slip cover to make a thin even layer.  Place on the microscope and focus.  Small black dots displaying Brownian motion will be easily observable.   It helps to cover the eye pieces of the microscope when operating in the video mode.

A substitute for the microspheres would be to use a dilute solution of milk on a slide.  The globules of fat will show Brownian motion. 

To do Brownian motion with smoke particles you will do the following procedure.  

NOTE: This is much more difficult than using the microspheres and requires constant attention and adjustment.   

Attach the barrel lens to the video camera and then the smoke cell attachment to the barrel lens.  Fill the smoke cell with smoke from a match and illuminate with either the laser or the Maglite.  This will show up fairly well on a TV if you are not very far away, although the contrast will still be fairly poor.  The best way to do this is to video capture a segment of the motion and apply some sharpening and false color, and then project the rendered video.

• Ziazhong Zeng, Mingzhen Shao, Xiaoqi Zeng, "Exploring Brownian Motion with a Millikan Oil Drop Apparatus", TPT, Vol. 62, #7, Oct. 2024, p. 579.
• Makito Miyazaki, Yosuke Yamazaki, Yamato Hasegawa, "Analysis of Brownian Motion by Elementary School Students", TPT, Vol. 60, #6, Sept. 2022, p. 478.
• Se-yuen Mak, "Brownian Motion Using a Laser Pointer", TPT, Vol. 36, # 6, Sept. 1998, p. 342. 
• Bill Reid, "Viewer for Brownian Motion", TPT, Vol. 29, # 1, Jan. 1991, p. 52. 
• Haym Kruglak, "Taking a Random Walk on TV", TPT, Vol. 26, # 3, Mar. 1988, p. 157.
• Sister Martha Ryder, "Brownian Movement", TPT, Vol. 12, # 9, Dec. 1974, p. 574.
• Azul Maria Brigante, Corina Revora, Gabriel Fernando Volonnino, Marcos Damian Perez, Gabriela Pasquini, Maria Gabriela Capeluto, "Experimentation on Stochastic Trajectories: From Brownian Motion to Inertial Confined Dynamics", AJP, Vol. 92, #4, April 2024, p. 280. 
• Karl D. Stephan, "An Economical Smoke Chamber and Light -Sheet Microscope System for Experiments in Fluid Dynamics and Electrostatics", AJP, Vol. 91, #4, April 2023, p. 316.
• Philip Pearle, Brian Collett, David Bilderback, Dara Newman, Scott Samuels, "What Brown Saw and You Can Too", AJP, Vol. 78, # 12, p. 1278, Dec. 2010.
• Joseph Peidle, Chris Stokes, Robert Hart, Melissa Franklin, Ronald Newburgh, Joon Pahk, Wolfgang Rueckner, Aravi Samuel, "Inexpensive Microscopy for Introductory Laboratory Courses", AJP, Vol. 77, #10, Oct. 2009, p.  931.
• Dongdong Jia, Jonathan Hamilton, Lenu M. Zaman, Anura Goonewardene, "The Time, Size, Viscosity, and Temperature Dependence of the Brownian Motion of Polystyrene Microspheres", AJP, Vol. 75, # 2, p. 111, Feb. 2007.
• Paul Nakroshis, Matthew Amoroso, Jason Legere, Christian Smith, "Measuring Boltzmann's Constant Using Video Microscopy of Brownian Motion", AJP, Vol. 71, # 6, June 2003, p. 568.
• Don S. Lemons and Anthony Gythiel, "Paul Langevin's 1908 Paper 'On the Theory of Brownian Motion'", AJP, Vol. 65, #11, Nov. 1997, p. 1079.
• Haym Kruglak, "Brownian Movement: An Improved TV Demonstration", AJP, Vol. 55, # 10, Oct. 1987, p. 955.
• Henry Unruh, Jr., Patrick M. Maxton, and Jonathan Schwartz, "Experimental Study of the Brownian Motion of a Harmonically Bound Particle", AJP, Vol. 47, #9, Sept. 1979, p. 827.
• Robert Stoller, "Viewing Brownian Motion With Laser Light", AJP, Vol. 44, #2, Feb. 1976, p. 188.
• George Barns, "A Brownian Motion Demonstration Using Television", AJP, Vol. 41, #2, Feb. 1973, p. 278.
• Noel A. Clark and Joseph H. Lunacek, "A Study of Brownian Motion Using Light Scattering", AJP, Vol. 37, #9, Sep. 1969, p. 853.
• Richard J. Fitzgerald, "Graphene Membranes’ Anomalous Dynamics", Physics Today, Vol. 69, #11, Nov. 2016, p. 24.
• Douglas J. Durian, "Ballistic Motion of a Brownian Particle", Physics Today, June 2015, p. 10.
• Mark G. Raizen, Tangcong Li, "Raizen and Li Reply", Physics Today, June. 2015, p. 11.
• Mark G. Raizen, Tangcong Li, "The Measurements Einstein Deemed Impossible", Physics Today, Jan. 2015, p. 56.
• H-16, Freier  & Anderson,   A Demonstration Handbook for Physics,  
• M- 223, Richard Manliffe Sutton,  Demonstration Experiments in Physics. 
• A-48, Richard Manliffe Sutton,  Demonstration Experiments in Physics,  p. 463.
• A-50, Richard Manliffe Sutton,  Demonstration Experiments in Physics,  p. 463
• A-49, Richard Manliffe Sutton,  Demonstration Experiments in Physics,  p. 463
• A-51, Richard Manliffe Sutton,  Demonstration Experiments in Physics,  p. 464
• Eli Barkai, Yuval Garini, Ralf Metzler, "Strange Kinetics of Single Molecules in Living Cells", Physics Today, Aug. 2012, p. 29.
• "An Optical Speed Trap for Brownian Motion", Physics Today, July 2010, p. 19.
• Wallace A. Hilton, "Brownian Motion", A Potpourri of Physics Teaching Ideas - Heat and Fluids, p. 116.
• R. W. Pohl, "The Free Movement of Molecules in a Liquid: The Brownian Movement", Physical Principles of Mechanics and Acoustics, p. 152.
• "Experiments with Camphor", The Boy Scientist, p. 150.
• Reese Salmon, Candace Robbins, Kyle Forinash, "Brownian Motion Using Video Capture", Eur. J. Phys. Vol. 23, 2002, p. 249.
• Kenneth Lonnquist, "Experiments in Quantum Mechanics - Brownian Motion", Physics Laboratory, Physics Department, Colorado State University. 
• Little Shop of Physics, "The Motion of Molecules", Colorado State University. 
• Richard E. Berg, "HINTS: Simplified Brownian Motion Demonstration", PIRA Newsletter, Vol. 3, # 5, September 10, 1988, p. 3.
• Ron Hipschman, "Brownian Motion (Molecular Buffering)", 1980 Exploratorium Cookbook II, Recipe # 128.
• "Brownian Movement Smoke Cell", EMD / A Division of Fisher Scientific, 92/93 Physics & Technology, p. 106.
• "Brownian Movement Apparatus", Central Scientific Company, 1990.
• "Instruction for Cenco Cat. # 71268  Brownian Movement Viewer", Central Scientific Company, 1990.
• "Brownian Movement", Central Scientific Company, 1960.
• A. Mason Turner, "E.M.E. Molecular Motion Demonstrator", Study Guide, 1975.
• Griffin & George, " Whitley Bay Smoke Cell XCU-300-T", Instructions.
• Ron Hipschman, "Brownian Motion (Molecular Buffering)", 1980 Exploratorium Cookbook II, Recipe # 127.
• Yaakov Kraftmakher, "7.20, Demonstrations with a Microscope", Experiments and Demonstrations in Physics, ISBN 981-256-602-3, p. 493.
• W. Bolton, "Brownian Motion", Book I - Properties of Materials, Physics Experiments and Projects, 1968, p. 30.
• Tap-L conversation with Gerald Zani, Aug 2005.
• Tap-L conversation with Sam Sampere, Sep 2007.

Disclaimer: These demonstrations are provided only for illustrative use by persons affiliated with The University of Iowa and only under the direction of a trained instructor or physicist.  The University of Iowa is not responsible for demonstrations performed by those using their own equipment or who choose to use this reference material for their own purpose.  The demonstrations included here are within the public domain and can be found in materials contained in libraries, bookstores, and through electronic sources.  Performing all or any portion of any of these demonstrations, with or without revisions not depicted here entails inherent risks.  These risks include, without limitation, bodily injury (and possibly death), including risks to health that may be temporary or permanent and that may exacerbate a pre-existing medical condition; and property loss or damage.  Anyone performing any part of these demonstrations, even with revisions, knowingly and voluntarily assumes all risks associated with them.

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Kinetic theory.

“Under the microscope one, to some extent, immediately sees a part of thermal energy in the form of mechanical energy of the moving particles.” —A. Einstein 1915

What it Shows

Tiny latex spheres in water, viewed under a microscope, undergo a kind of random jiggling motion called Brownian motion —named after the botanist Robert Brown , who observed this kind of motion in 1827 when looking at tiny pollen grains. The spheres are all 1.054 micron in diameter. Each particle can be seen...

Drum evacuated by vacuum pump; crushed by atmospheric bombardment.

What it shows:

With an air pressure of 10 5 Nm -2 at sea level, even a heavy duty oil drum will be crushed if it has nothing inside to balance the pressure.

How it works:

The screw cap on the drum is fitted with a vacuum pump connector. Simply turn on the pump and wait; it takes about 8 minutes to pump down, so you can carry on with what you were doing interrupted by various creaks and bangs as the drum's side walls begin to give....

Smoke cell under microscope; smoke particles seen bombarded by air molecules.

Brownian motion shows direct evidence of the incessant motion of matter due to thermal energy. Here we use the random bombardment of smoke particles by air molecules.

How it Works

The CENCO Brownian Movement Apparatus consists of a metal chamber with a glass viewing window on top and a lens on one side (see figure 1). Smoke from a piece of smoldering rope or match is drawn into the chamber through an inlet tube by squeezing the rubber bulb....

Simulation of molecular motion (Brownian, diffusion, etc.) with ball bearings on shaking table.

Two dimensional simulations of molecular dynamics and crystal structure using ball bearings. It can be used to show qualitatively the dynamics of liquids and gases, and illustrate crystalline forms and dislocations.

The molecular dynamics simulator is more commonly known as a shaking table. It consists primarily of a circular shallow walled glass table that is oscillated vertically so as to vibrate and...

CO 2 and He balloons in liquid nitrogen.

Cooling a gas causes a proportional decrease in volume with the drop in absolute temperature. A gas such as helium, which remains close to ideal at low temperatures, shows a four-fold decrease in volume when taken from room temperature 330K to liquid nitrogen temperature, 77K. Carbon dioxide however, sublimes at 194.5K, so is solid at 77K. Oxygen liquefies at 90K (S.T.P.). A qualitative demonstration of these effects can be shown with gas filled balloons.

How it works:...

When evacuated, held together by bombardment of atmospheric molecules.

Two brass hemispheres are brought together and evacuated, and are held together by the pressure of the atmosphere.

Two brass hemispheres fit together to form an air-tight seal. One has a vacuum pump attachment and stop cock; the completed sphere can evacuated using a vacuum pump under a minute. As atmospheric pressure is 10 5 Nm -2 , the 11cm diameter hemispheres are held together by a force of 15000N. Invite members of your...

A model of molecular motion and pressure using practice golf balls.

The kinetic energy of gas molecules bouncing off a surface causes pressure.

Increasing the molecules' speeds increases the pressure and the volume of the gas.

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Brownian Motion

Students observe the random movement of particles in air.

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• Smoke Cell (AKA: Whitley Bay Smoke Cell)

Comprehensive Modeling 1 / f 1 𝑓 1/f 1 / italic_f Noises in TRNGs with Fractional Brownian Motion

Building upon the foundational work of atomic clock physicists Barnes and Allan, this paper presents a highly scalable and numerically exact framework for modeling 1 / f 1 𝑓 1/f 1 / italic_f noise in oscillatory True Random Number Generators (TRNGs) and assessing their cryptographic security. By employing Fractional Brownian Motion, the framework constructs Gaussian non-stationary processes that represent these noise spectra accurately and in a mathematically sound way. Furthermore, it establishes several critical properties, including optimal bounds on the achievable generation rate of cryptographically secure bits.

1 Introduction

State-of-the-art random number generators leverage physical phenomena  [ SK14 ] such as noise in electronic circuits [ BLMT11 ] . However, modeling such noises is challenging due to the complex nature of low-frequency noise with 1 / f α 1 superscript 𝑓 𝛼 1/f^{\alpha} 1 / italic_f start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT spectral law. Recent studies have made significant progress by introducing simulation and Monte Carlo inference frameworks [ PV24 , BCF + 24 ] . Nevertheless, the fundamental challenge remains: the development of a rigorous mathematical foundation, including a formal spectral law for non-stationary 1 / f 1 𝑓 1/f 1 / italic_f noise, and the creation of highly efficient numerical inference methods.

This work revisits the established flicker noise model introduced by atomic clock physicists Barnes and Allan [ BA66 , Bar71 , BJTA82 ] , proposing to model all 1 / f α 1 superscript 𝑓 𝛼 1/f^{\alpha} 1 / italic_f start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT noises with − 1 ≤ α ≤ 0 1 𝛼 0 -1\leq\alpha\leq 0 - 1 ≤ italic_α ≤ 0 (encompassing flicker and thermal noise) using Fractional Brownian Motion as defined through Riemann-Liouville fractional integration [ SL95 ] . The spectral power law is further justified using the concept of the time-varying Wigner-Ville spectrum. Unlike previous works [ PV24 , BCF + 24 ] , this study achieves numerically exact results with a highly scalable computation method.

Throughout the remainder of this paper, we analyze a noisy oscillator characterised by its stochastically-perturbed phase:

where the phase excess ξ ⁢ ( t ) 𝜉 𝑡 \xi(t) italic_ξ ( italic_t ) is modelled as a stochastic (Gaussian) process. Bits are sampled using a 2 ⁢ π 2 𝜋 2\pi 2 italic_π -periodic square wave, which may have an imperfect duty cycle:

The frequency, f ⁢ ( t ) 𝑓 𝑡 f(t) italic_f ( italic_t ) , is defined as the derivative of the phase:

The relative frequency deviation, Y ⁢ ( t ) 𝑌 𝑡 Y(t) italic_Y ( italic_t ) , is expressed as  [ HAB81 ] :

This deviation follows the empirically verified power spectral density  [ HAB81 ] :

where h α subscript ℎ 𝛼 h_{\alpha} italic_h start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT are constants dependent on the hardware. For flicker frequency modulation (flicker-FM), α = − 1 𝛼 1 \alpha=-1 italic_α = - 1 , while for the thermal noise component, α = 0 𝛼 0 \alpha=0 italic_α = 0 . In practical scenarios, we typically consider α ∈ { − 1 , 0 } 𝛼 1 0 \alpha\in\{-1,0\} italic_α ∈ { - 1 , 0 } .

The goal of this paper is to present a comprehensive and rigorous model for the phase Φ ⁢ ( t ) Φ 𝑡 \Phi(t) roman_Φ ( italic_t ) , including justification of its spectral properties, find a link to hardware-dependent constants h α subscript ℎ 𝛼 h_{\alpha} italic_h start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , establish security guarantees on the entropy rate of generated bits b ⁢ ( t ) 𝑏 𝑡 b(t) italic_b ( italic_t ) .

2 Preliminaries

Definition 2.1 (riemann-liouville fractional brownian motion) ..

The Fractional Brownian Motion is defined as

where B u subscript 𝐵 𝑢 B_{u} italic_B start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT is the standard Brownian motion.

In our model, the flicker frequency modulation will correspond to H = 1 𝐻 1 H=1 italic_H = 1 and white noise to H = 1 2 𝐻 1 2 H=\frac{1}{2} italic_H = divide start_ARG 1 end_ARG start_ARG 2 end_ARG , because of the matching spectral properties (as shall be demonstrated later). Note that there are alternative ways of defining the fractional Brownian Motions, particularly due to Mandelbrot-Van Ness  [ MVN68 ] , but as opposed to the Riemann-Liouville variant, their spectral laws don’t account for the important 1 / f 1 𝑓 1/f 1 / italic_f flicker noise  [ SL95 ] .

Definition 2.2 (Gauss Hypergeometric Function) .

The hypergeometric function is defined for | z | < 1 𝑧 1 |z|<1 | italic_z | < 1 by

Definition 2.3 (Gaussian Process) .

A gaussian process is completely determined by its covariance and mean functions. Given the covariance and mean functions, sampling can be done with Cholesky’s method  [ WB24 ] .

Definition 2.4 (Schur Complement) .

Given a block matrix M = [ A B C D ] 𝑀 matrix 𝐴 𝐵 𝐶 𝐷 M=\begin{bmatrix}A&B\\ C&D\end{bmatrix} italic_M = [ start_ARG start_ROW start_CELL italic_A end_CELL start_CELL italic_B end_CELL end_ROW start_ROW start_CELL italic_C end_CELL start_CELL italic_D end_CELL end_ROW end_ARG ] , the Schur complement of D in M is written M ∖ D = A − B ⁢ D − 1 ⁢ C 𝑀 𝐷 𝐴 𝐵 superscript 𝐷 1 𝐶 M\setminus D=A-BD^{-1}C italic_M ∖ italic_D = italic_A - italic_B italic_D start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_C .

Proposition 2.5 (Conditional Gaussian Distribution  [ Cot74 ] ) .

subscript 𝜇 𝑋 subscript Σ 𝑋 𝑌 superscript subscript Σ 𝑌 1 𝑦 subscript 𝜇 𝑌 \mu^{\prime}=\mu_{X}+\Sigma_{XY}\Sigma_{Y}^{-1}(y-\mu_{Y}) italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_μ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT + roman_Σ start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_y - italic_μ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) and variance Σ ′ = Σ X − Σ X ⁢ Y ⁢ Σ Y − 1 ⁢ Σ Y ⁢ X = Σ ∖ Σ Y superscript Σ ′ subscript Σ 𝑋 subscript Σ 𝑋 𝑌 superscript subscript Σ 𝑌 1 subscript Σ 𝑌 𝑋 Σ subscript Σ 𝑌 \Sigma^{\prime}=\Sigma_{X}-\Sigma_{XY}\Sigma_{Y}^{-1}\Sigma_{YX}=\Sigma% \setminus\Sigma_{Y} roman_Σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_Σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT - roman_Σ start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_Σ start_POSTSUBSCRIPT italic_Y italic_X end_POSTSUBSCRIPT = roman_Σ ∖ roman_Σ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT .

Corollary 2.6 (Variance of Conditional Gaussian by Schur Completion) .

The variance Σ ′ superscript Σ ′ \Sigma^{\prime} roman_Σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT of the conditional distribution X 1 | X 2 conditional subscript 𝑋 1 subscript 𝑋 2 X_{1}|X_{2} italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of jointly gaussian distributions can be expressed as Σ ′ = Σ X ∖ Σ Y superscript Σ ′ subscript Σ 𝑋 subscript Σ 𝑌 \Sigma^{\prime}=\Sigma_{X}\setminus\Sigma_{Y} roman_Σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_Σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ∖ roman_Σ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT .

Wigner-Ville transform is known to be a very powerful tool for signal processing in the time frequency plane [ MF85 ] . It allows us to correctly extend the spectrum to non-stationary processes (alternatively, one can use instantenous spectra  [ Pri67 ] ).

Definition 2.7 (Wigner-Ville Spectrum) .

For a stochastic process X ⁢ ( t ) 𝑋 𝑡 X(t) italic_X ( italic_t ) with covariance function C X subscript 𝐶 𝑋 C_{X} italic_C start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT we define

which reduces to the standard notion of spectrum (Einstein-Wiener-Khinchin Theorem) when X ⁢ ( t ) 𝑋 𝑡 X(t) italic_X ( italic_t ) is (wide-sense) stationary.

Example 2.8 .

For the Brownian Motion one obtains W X ⁢ ( t , f ) = 2 ⁢ sin 2 ⁡ ( t ⁢ ω ) / ω 2 subscript 𝑊 𝑋 𝑡 𝑓 2 superscript 2 𝑡 𝜔 superscript 𝜔 2 W_{X}(t,f)=2\sin^{2}(t\omega)/\omega^{2} italic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_t , italic_f ) = 2 roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_t italic_ω ) / italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT where ω = 2 ⁢ π ⁢ f 𝜔 2 𝜋 𝑓 \omega=2\pi f italic_ω = 2 italic_π italic_f is the angular frequency.

To derive closed-form entropy formulas, we will need the machinery of elliptic functions, with popular definitions see [ DLMF , (20.2.3) ]

Definition 2.9 (Elliptic Functions) .

The third elliptic theta function is defined as

or with parametrization q = e 𝐢 ⁢ π ⁢ τ 𝑞 superscript e 𝐢 𝜋 𝜏 q=\mathrm{e}^{\mathbf{i}\pi\tau} italic_q = roman_e start_POSTSUPERSCRIPT bold_i italic_π italic_τ end_POSTSUPERSCRIPT :

For completeness, wre recall the definition of min-entropy which will be used to evaluate security of output bits.

Definition 2.10 (Min-Entropy) .

We assume that the phase excess is a Gaussian process described by a combination of fractional Brownian Motions. Namely,

for the appropriate choice of H 𝐻 H italic_H . The phase Φ ⁢ ( t ) Φ 𝑡 \Phi(t) roman_Φ ( italic_t ) has the same covariance matrix, but a linearly growing mean.

3.1 Covariance Characterization of fBMs

We first establish basic properties of fractional Brownian Motions.

Theorem 3.1 (Hypergeometric Formulas for Covariance of fBMs) .

For any H > 0 𝐻 0 H>0 italic_H > 0 we have

In particular, for H = 1 𝐻 1 H=1 italic_H = 1 (flicker noise)

and for H = 1 / 2 𝐻 1 2 H=1/2 italic_H = 1 / 2 (white noise)

This family has good stochastic properties, in particular it is semistable for all H > 0 𝐻 0 H>0 italic_H > 0 and self-affine with the scaling exponent H 𝐻 H italic_H . Naming the kernels by "flicker" and "white" will be justified in the next section, by establishing their spectral properties. Figure   1 below shows samples from fractional Brownian Motions.

Corollary 3.2 (Self-affinity) .

For any a > 0 𝑎 0 a>0 italic_a > 0 we have L H ⁢ ( t ) ⁢ = 𝑑 ⁢ a − H ⁢ L H ⁢ ( a ⁢ t ) subscript 𝐿 𝐻 𝑡 𝑑 superscript 𝑎 𝐻 subscript 𝐿 𝐻 𝑎 𝑡 L_{H}(t)\overset{d}{=}a^{-H}L_{H}(at) italic_L start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_t ) overitalic_d start_ARG = end_ARG italic_a start_POSTSUPERSCRIPT - italic_H end_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_a italic_t ) .

Corollary 3.3 (Variances) .

𝑡 𝜏 𝑡 K_{w}(t,t+\tau)=t italic_K start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_t , italic_t + italic_τ ) = italic_t respectively.

The result below approximates a flicker covariance when the lag is much smaller than the evaluation time (that is, τ ≪ t much-less-than 𝜏 𝑡 \tau\ll t italic_τ ≪ italic_t ).

Corollary 3.4 (Reparametrization of Flicker Covariance) .

In particular, for small ρ 𝜌 \rho italic_ρ we get

3.2 Spectral Properties of Phase

We start by establishing the desired spectral properties of f 𝑓 f italic_f -noises. To properly handle non-stationarity, we use time-varying spectrum   [ Fla89 , SL95 ] .

Proposition 3.5 (Spectrum of Fractional Brownian Motion  [ SL95 ] ) .

For t ⁢ ω ≫ 1 much-greater-than 𝑡 𝜔 1 t\omega\gg 1 italic_t italic_ω ≫ 1 ,

where ω = 2 ⁢ π ⁢ f 𝜔 2 𝜋 𝑓 \omega=2\pi f italic_ω = 2 italic_π italic_f is the angular frequency and S 𝑆 S italic_S is understood as the time-averaged Wigner-Ville spectrum.

Note that the above result justifies the choice H = 1 / 2 𝐻 1 2 H=1/2 italic_H = 1 / 2 and H = 1 𝐻 1 H=1 italic_H = 1 for modelling white and flicker frequency modulations. Now, by Equation   4 and properties of the Fourier transform we obtain

2 𝐻 1 superscript subscript 𝑓 0 2 superscript 𝑓 𝛼 2 c_{H}=(2\pi)^{2H+1}f_{0}^{2}f^{\alpha-2} italic_c start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = ( 2 italic_π ) start_POSTSUPERSCRIPT 2 italic_H + 1 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_α - 2 end_POSTSUPERSCRIPT and α = 1 − 2 ⁢ H 𝛼 1 2 𝐻 \alpha=1-2H italic_α = 1 - 2 italic_H . Limiting out attention to α ∈ { − 1 , 0 } 𝛼 1 0 \alpha\in\{-1,0\} italic_α ∈ { - 1 , 0 } , we obtain the following important result

Corollary 3.6 (Phase Kernel with Noise-Level Constants) .

The phase excess ( 11 ) is a guassian process with explicit form given by

Furthermore, the phase is a guassian process with covariance

This establishes the promised link of our model to hardware-dependent constants determining noise levels.

3.3 Periodic Gaussian Distribution

While we model the absolute phase excess, whereas sampling is done through a periodic square-wave. To transfer the probability calculations to the bit space, we establish an auxiliary result of independent interest that gives an exact density for a periodically wrapped Gaussian distribution.

Theorem 3.7 .

Suppose that X ∼ Norm ⁢ ( μ , σ 2 ) similar-to 𝑋 Norm 𝜇 superscript 𝜎 2 X\sim\mathrm{Norm}(\mu,\sigma^{2}) italic_X ∼ roman_Norm ( italic_μ , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . Then Y = X mod 1 𝑌 modulo 𝑋 1 Y=X\bmod 1 italic_Y = italic_X roman_mod 1 have the density

and Y = X mod 2 ⁢ π 𝑌 modulo 𝑋 2 𝜋 Y=X\bmod 2\pi italic_Y = italic_X roman_mod 2 italic_π has the density

where ϑ 3 subscript italic-ϑ 3 \vartheta_{3} italic_ϑ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT denotes the third Jacobi elliptic theta function.

Remark 3.8 .

The expression provides a very quickly convergent series. Numerical evaluation is easy with modern mathematical software.

This result is interesting on its own, but will apply it to establish the security of bits sampled through observing low and high positions of any gaussian phase

3.4 Conditional Bias and Min-Entropy of Sampled Bits

Theorem 3.9 ..

subscript italic-ϕ 0 𝜇 𝑡 \mu(t)=\phi_{0}+\mu t italic_μ ( italic_t ) = italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_μ italic_t and is measured at times t k subscript 𝑡 𝑘 t_{k} italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT for k = 1 ⁢ … ⁢ n 𝑘 1 … 𝑛 k=1\ldots n italic_k = 1 … italic_n . Let

be the leftover variance, given the history of the previous phase values Φ ⩽ n − 1 = ( Φ t k ) k < n subscript Φ absent 𝑛 1 subscript subscript Φ subscript 𝑡 𝑘 𝑘 𝑛 \Phi_{\leqslant n-1}=(\Phi_{t_{k}})_{k<n} roman_Φ start_POSTSUBSCRIPT ⩽ italic_n - 1 end_POSTSUBSCRIPT = ( roman_Φ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k < italic_n end_POSTSUBSCRIPT . Let α 𝛼 \alpha italic_α be the duty cycle. Then

Corollary 3.10 (Output Conditional Min-Entropy) .

The conditional min-entropy of generated bits, under the security model where an attacker can know the previous phase locations, equals

where the left-over variance σ 2 superscript 𝜎 2 \sigma^{2} italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is evaluated using Schur complement:

Notice that the growth of min-entropy is unimodal when the duty cycle is imbalanced; it increases up to 1.0 and then decreases, stabilizing at a lower value. This observation underscores the necessity for precise numerical analysis of the model, especially in the context of future post-processing.

3.5 Exact Results for Flicker Noises

Suppose that bits are sampled at times t k = k ⁢ τ subscript 𝑡 𝑘 𝑘 𝜏 t_{k}=k\tau italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_k italic_τ . Consider the leftover variance:

Below we observe that the leftover variance remains bounded, which will ensure enough entropy, through large enough variance, for security of the sampled bits.

Corollary 3.11 (Bounded Leakage Property of Flicker Phase Noise) .

For any Gaussian phase process Φ Φ \Phi roman_Φ with flicker covariance K Φ = C ⋅ K f subscript 𝐾 Φ ⋅ 𝐶 subscript 𝐾 𝑓 K_{\Phi}=C\cdot K_{f} italic_K start_POSTSUBSCRIPT roman_Φ end_POSTSUBSCRIPT = italic_C ⋅ italic_K start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , the leftover variance increases and obeys the bound

With the leakage under control, we are ready to conclude the formulas for the bandwith

Corollary 3.12 (Sampling Rate under Flicker Noise) .

The sampling window τ 𝜏 \tau italic_τ should satisfy

equivalently the bit rate should be at most

where the exact constant depends on the targeted security level.

Remark 3.13 .

This result is optimal and somewhat pessimistic - flicker noise accumulates quicker than thermal noise, but also leaks significantly. With typical hardware constants one can improve upon the rate offered by the thermal noise, albeit not by many orders of magnitude.

Example 3.14 .

Suppose that f 0 = 50 ⁢ M ⁢ H ⁢ z subscript 𝑓 0 50 M H z f_{0}=50\mathrm{MHz} italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 50 roman_M roman_H roman_z and h − 1 = 10 − 10 subscript ℎ 1 superscript 10 10 h_{-1}=10^{-10} italic_h start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 10 end_POSTSUPERSCRIPT . With Corollary   3.10 , Corollary   3.11 , and Corollary   3.6 we find that for min-entropy of at least 0.99 0.99 0.99 0.99 we need τ ⩾ 2 ⋅ 10 − 4 𝜏 ⋅ 2 superscript 10 4 \tau\geqslant 2\cdot 10^{-4} italic_τ ⩾ 2 ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT , giving the sampling rate of 5 ⋅ 10 3 ⁢ bps ⋅ 5 superscript 10 3 bps 5\cdot 10^{3}\mathrm{bps} 5 ⋅ 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_bps .

3.6 Measuring Flicker Noise through Alan Variance

For a signal Y ⁢ ( y ) 𝑌 𝑦 Y(y) italic_Y ( italic_y ) , one defines the Allan variance with lag τ 𝜏 \tau italic_τ as

The following result connects the Allan variance to the noise level, effectively giving a way of measuring hardware-dependent constants in the stochastic model.

Proposition 3.15 .

The Allan variance of any process Y 𝑌 Y italic_Y with covariance function K 𝐾 K italic_K equals

Using Proposition   3.15 we now derive an important characterization of the Allan variance in terms of noise-level constants. Estimating the Allan variance and inverting this relation will allow us to recover the hardware constant h − 1 subscript ℎ 1 h_{-1} italic_h start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT .

Theorem 3.16 (Allan Variance of Flicker Phase Fluctuations) .

Suppose that Φ ⁢ ( t ) Φ 𝑡 \Phi(t) roman_Φ ( italic_t ) follows a gaussian process with the flicker covariance matrix C ⋅ K f ⋅ 𝐶 subscript 𝐾 𝑓 C\cdot K_{f} italic_C ⋅ italic_K start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT . Then

for τ / t = O ⁢ ( 1 ) 𝜏 𝑡 𝑂 1 \tau/t=O(1) italic_τ / italic_t = italic_O ( 1 ) .

By combining this with Corollary   3.6 , we obtain the following important result that allows to retrieve the flicker constant from estimated Allan variance:

Corollary 3.17 (Flicker Constant from Allan Variance) .

4 experimental evaluation, 4.1 hardware experiment.

The setup consists of two ring oscillators with frequencies 51.397 MHz and 52.403 MHz, the first one sampling bits as the low/high state of the second ring.

Using oscilloscope, about 1M of subsequent phase locations were retrieved. The Allan variance was estimated as the running average over this single long path. The results are shown in Figure   5 ; note that the running average captures transition from lower levels of Allan variance (thermal noise) to the stable higher level (corresponding to flicker).

By Corollary   3.17 , where τ = 1 / f 0 = 1 / 51.397 ⁢ s 𝜏 1 subscript 𝑓 0 1 51.397 𝑠 \tau=1/f_{0}=1/51.397s italic_τ = 1 / italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / 51.397 italic_s , we obtained the level of flicker noise

the magnitude aligning with the literature.

It remains to choose an appropriate sampling rate, to achieve enough leftover variance. By Corollary   3.10 , for min-entropy of 0.99 0.99 0.99 0.99 the leftover variance should be at least 3.4 3.4 3.4 3.4 . By Corollary   3.11 , and Corollary   3.6 we finally find that

is the secure sampling rate. In this experiment, we ignore the smaller contribution from the thermal noise - although it’s possible to study the mixture with the discussed framework.

4.2 Software Implementation

For computation on gaussian kernels, this paper uses the 𝖦𝖯𝖥𝗅𝗈𝗐 𝖦𝖯𝖥𝗅𝗈𝗐 \mathsf{GPFlow} sansserif_GPFlow machine-learning library  [ MvN + 17 ] . For instance, the code snippet below shows an implementation of the flicker kernel (up to a constant).

[linenos, frame=single, breaklines=true, breakanywhere=true]python def K_f(x, y=None): if y is None: y = x.T t = tf.minimum(x, y) tau = tf.maximum(x, y) - t cov = tf.sqrt(t * (t + tau)) * (2 * t + tau)  - tf.square(tau) * tf.atanh(tf.sqrt(t) / tf.sqrt(t + tau + tf.keras.backend.epsilon())) cov = 0.25 * cov return cov

5.1 Proof of Theorem   3.1

It is easy to see that that the autocorrelation function is

The first formula follows now by the integral identity for hypergeometric functions; the simpler formula for H = 1 𝐻 1 H=1 italic_H = 1 is the consequence of the identity

as well as Γ ⁢ ( 3 / 2 ) = π / 2 Γ 3 2 𝜋 2 \Gamma(3/2)=\sqrt{\pi}/2 roman_Γ ( 3 / 2 ) = square-root start_ARG italic_π end_ARG / 2 .

5.2 Proof of Theorem   3.7

We first derive the density f ⁢ ( x ) 𝑓 𝑥 f(x) italic_f ( italic_x ) of Y = X mod 1 𝑌 modulo 𝑋 1 Y=X\bmod 1 italic_Y = italic_X roman_mod 1 when X 𝑋 X italic_X is centered, i.e., μ = 0 𝜇 0 \mu=0 italic_μ = 0 . For a fixed x 𝑥 x italic_x , we have

Define τ = 1 2 ⁢ π ⁢ σ 2 ⋅ 𝐢 𝜏 ⋅ 1 2 𝜋 superscript 𝜎 2 𝐢 \tau=\frac{1}{2\pi\sigma^{2}}\cdot\mathbf{i} italic_τ = divide start_ARG 1 end_ARG start_ARG 2 italic_π italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ⋅ bold_i and z = − x 2 ⁢ σ 2 ⋅ 𝐢 𝑧 ⋅ 𝑥 2 superscript 𝜎 2 𝐢 z=-\frac{x}{2\sigma^{2}}\cdot\mathbf{i} italic_z = - divide start_ARG italic_x end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ⋅ bold_i . Using the identity

and the definition of the theta function, we can express this as

Utilizing the Jacobi identities, specifically

To generalize to a non-centered distribution with μ ≠ 0 𝜇 0 \mu\neq 0 italic_μ ≠ 0 , substitute x 𝑥 x italic_x with x − μ 𝑥 𝜇 x-\mu italic_x - italic_μ . To derive the density for Y mod 2 ⁢ π modulo 𝑌 2 𝜋 Y\bmod 2\pi italic_Y roman_mod 2 italic_π , consider Y = Y / ( 2 ⁢ π ) 𝑌 𝑌 2 𝜋 Y=Y/(2\pi) italic_Y = italic_Y / ( 2 italic_π ) , replacing x 𝑥 x italic_x with x / ( 2 ⁢ π ) 𝑥 2 𝜋 x/(2\pi) italic_x / ( 2 italic_π ) , μ 𝜇 \mu italic_μ with μ / ( 2 ⁢ π ) 𝜇 2 𝜋 \mu/(2\pi) italic_μ / ( 2 italic_π ) , and σ 2 superscript 𝜎 2 \sigma^{2} italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT with σ 2 / ( 2 ⁢ π ) 2 superscript 𝜎 2 superscript 2 𝜋 2 \sigma^{2}/(2\pi)^{2} italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

5.3 Proof of Theorem   3.9